1_MC1 ._. 
Maxwell_NP1 's_GE equations_NN2 One_MC1 of_IO the_AT chief_JJ peculiarities_NN2 of_IO this_DD1 treatise_NN1 is_VBZ the_AT doctrine_NN1 which_DDQ asserts_VVZ ,_, that_CST the_AT true_JJ electric_JJ current_JJ on_II which_DDQ the_AT electromagnetic_JJ phenomena_NN2 depend_VV0 ,_, is_VBZ not_XX the_AT same_DA thing_NN1 as_CSA the_AT current_NN1 of_IO conduction_NN1 ,_, but_CCB that_CST the_AT time_NNT1 variation_NN1 of_IO the_AT electric_JJ displacement_NN1 must_VM be_VBI taken_VVN into_II account_NN1 in_II estimating_VVG the_AT total_JJ movement_NN1 of_IO electricity_NN1 ._. 
JAMES_NP1 CLARK_NP1 MAXWELL_NP1 A_ZZ1 treatise_VV0 on_II electricity_NN1 and_CC magnetism_NN1 Oxford_NP1 1873_MC ALL_DB the_AT problems_NN2 we_PPIS2 shall_VM be_VBI concerned_JJ with_IW may_VM be_VBI solved_VVN by_II calling_VVG upon_II one_MC1 or_CC more_DAR of_IO the_AT following_JJ equations_NN2 :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
Where_RRQ do_VD0 the_AT above_JJ equations_NN2 come_VV0 from_II ?_? 
They_PPHS2 are_VBR contained_VVN (_( though_CS not_XX quite_RR in_II the_AT same_DA form_NN1 and_CC not_XX in_II the_AT same_DA system_NN1 of_IO units_NN2 )_) in_II Chapter_NN1 IX_MC of_IO Maxwell_NP1 's_GE Treatise_NN1 on_II electricity_NN1 and_CC magnetism_NN1 published_VVN in_II 1873_MC ._. 
Are_VBR we_PPIS2 to_TO conclude_VVI that_DD1 electromagnetic_JJ theory_NN1 has_VHZ made_VVN no_AT advance_NN1 in_II the_AT course_NN1 of_IO a_AT1 century_NNT1 ?_? 
That_DD1 conclusion_NN1 would_VM essentially_RR be_VBI correct_JJ ._. 
Our_APPGE technique_NN1 of_IO solving_VVG the_AT above_JJ equations_NN2 has_VHZ improved_VVN ,_, and_CC of_RR21 course_RR22 we_PPIS2 are_VBR in_II a_AT1 much_RR better_JJR position_NN1 now_RT to_TO evaluate_VVI the_AT material_NN1 constants_NN2 ,_, but_CCB fundamentally_RR electromagnetic_JJ theory_NN1 stands_VVZ now_RT as_CSA it_PPH1 stood_VVD a_AT1 century_NNT1 ago_RA ._. 
As_CS31 far_CS32 as_CS33 the_AT interrelationship_NN1 of_IO electromagnetic_JJ quantities_NN2 is_VBZ concerned_JJ Maxwell_NP1 knew_VVD as_RG much_DA1 as_CSA we_PPIS2 do_VD0 today_RT ._. 
He_PPHS1 did_VDD not_XX actually_RR suggest_VVI communication_NN1 between_II continents_NN2 with_IW the_AT aid_NN1 of_IO geostationary_JJ satellites_NN2 ,_, but_CCB if_CS he_PPHS1 was_VBDZ taken_VVN now_RT to_II a_AT1 satellite_NN1 ground-station_NN1 he_PPHS1 would_VM not_XX be_VBI numbed_VVN with_IW astonishment_NN1 ._. 
If_CS we_PPIS2 would_VM give_VVI him_PPHO1 half_DB an_AT1 hour_NNT1 to_TO get_VVI over_II the_AT shock_NN1 of_IO his_APPGE resurrection_NN1 he_PPHS1 would_VM quietly_RR sit_VVI down_RP with_IW a_AT1 piece_NN1 of_IO paper_NN1 (_( the_AT back_NN1 of_IO a_AT1 bigger_JJR envelope_NN1 ,_, I_PPIS1 suppose_VV0 )_) and_CC would_VM work_VVI out_RP the_AT relevant_JJ design_NN1 formulae_NN2 ._. 
The_AT lack_NN1 of_IO advance_NN1 on_II our_APPGE part_NN1 should_VM not_XX be_VBI attributed_VVN to_II the_AT idleness_NN1 of_IO a_AT1 century_NNT1 ,_, much_RR rather_RG to_II the_AT genius_NN1 of_IO Maxwell_NP1 ._. 
The_AT moment_NN1 he_PPHS1 conceived_VVD the_AT idea_NN1 of_IO the_AT displacement_NN1 current_NN1 ,_, a_AT1 new_JJ era_NN1 started_VVD in_II the_AT history_NN1 of_IO mankind_NN1 ._. 
Events_NN2 of_IO similar_JJ importance_NN1 did_VDD not_XX occur_VVI often_RR ._. 
Newton_NP1 's_GE Principia_FW and_CC Einstein_NP1 's_GE first_MD paper_NN1 on_II relativity_NN1 would_VM qualify_VVI ,_, and_CC perhaps_RR two_MC or_CC three_MC more_DAR learned_JJ papers_NN2 ,_, but_CCB that_DD1 's_VBZ about_II all_DB ._. 
If_CS we_PPIS2 assume_VV0 that_CST our_APPGE kind_NN1 of_IO beings_NN2 will_VM still_RR be_VBI around_II a_AT1 few_DA2 millennia_NNT2 hence_RR ,_, I_PPIS1 feel_VV0 certain_JJ that_CST the_AT nineteenth_MD century_NNT1 will_VM mainly_RR be_VBI remembered_VVN as_II the_AT century_NNT1 when_RRQ Maxwell_NP1 formulated_VVD his_APPGE equations_NN2 ._. 
What_DDQ was_VBDZ so_RG extraordinary_JJ about_II Maxwell_NP1 's_GE contribution_NN1 ?_? 
It_PPH1 was_VBDZ the_AT first_MD (_( and_CC may_VM be_VBI the_AT best_JJT )_) example_NN1 of_IO reaching_VVG a_AT1 synthesis_NN1 on_II the_AT basis_NN1 of_IO experimental_JJ evidence_NN1 ,_, mathematical_JJ intuition_NN1 ,_, and_CC prophetic_JJ insight_NN1 ._. 
The_AT term_NN1 &lsqb;_( formula_NN1 &rsqb;_) (_( 1.1_MC )_) had_VHD no_AT experimental_JJ basis_NN1 at_II the_AT time_NNT1 ._. 
By_II adding_VVG this_DD1 new_JJ term_NN1 to_II the_AT known_JJ equations_NN2 he_PPHS1 managed_VVD to_TO describe_VVI all_DB macroscopic_JJ electromagnetic_JJ phenomena_NN2 ._. 
And_CC when_RRQ relativity_NN1 came_VVD ,_, Newton_NP1 's_GE equations_NN2 were_VBDR found_VVN wanting_JJ but_CCB not_XX Maxwell_NP1 's_GE ;_; they_PPHS2 needed_VVD no_AT relativistic_JJ correction_NN1 ._. 
I_PPIS1 could_VM go_VVI on_RP for_IF a_AT1 long_JJ time_NNT1 in_II praise_NN1 of_IO Maxwell_NP1 ._. 
Unfortunately_RR we_PPIS2 have_VH0 little_DA1 time_NNT1 for_IF digressions_NN2 however_RRQV entertaining_JJ they_PPHS2 might_VM be_VBI ._. 
But_CCB before_CS we_PPIS2 get_VV0 down_RP to_II the_AT equations_NN2 I_PPIS1 must_VM say_VVI a_AT1 few_DA2 words_NN2 in_II31 defence_II32 of_II33 the_AT approach_NN1 I_PPIS1 choose_VV0 ._. 
I_PPIS1 know_VV0 it_PPH1 must_VM be_VBI hard_JJ for_IF anyone_PN1 to_TO accept_VVI a_AT1 set_NN1 of_IO equations_NN2 without_IW going_VVG through_II the_AT usual_JJ routine_NN1 of_IO presenting_VVG the_AT relevant_JJ experimental_JJ justifications_NN2 ._. 
It_PPH1 might_VM seem_VVI a_RR21 little_RR22 unreasonable_JJ at_II first_MD sight_NN1 but_CCB believe_VV0 me_PPIO1 this_DD1 is_VBZ a_AT1 possible_JJ approach_NN1 ,_, and_CC under_II the_AT circumstances_NN2 it_PPH1 may_VM very_RG well_RR be_VBI the_AT best_JJT approach_NN1 ._. 
You_PPY are_VBR already_RR familiar_JJ with_IW the_AT mathematical_JJ operations_NN2 curl_VV0 ,_, div_MC ,_, and_CC grad_NN1 (_( I_PPIS1 prefer_VV0 using_VVG them_PPHO2 in_II vector-operator_JJ form_NN1 )_) ,_, and_CC you_PPY need_VV0 no_AT introduction_NN1 to_II the_AT concepts_NN2 of_IO electricity_NN1 ._. 
You_PPY have_VH0 heard_VVN about_II electric_JJ charge_NN1 ,_, current_JJ ,_, magnetic_JJ field_NN1 ,_, etc_RA ._. 
All_DB the_AT eqns_NN2 (_( 1.1_MC )_) -(1.7)_JJ do_VD0 is_VBZ go_NN1 give_VV0 a_AT1 number_NN1 of_IO relationships_NN2 between_II these_DD2 quantities_NN2 ._. 
So_RR if_CS some_DD of_IO them_PPHO2 are_VBR known_VVN ,_, you_PPY can_VM use_VVI the_AT equations_NN2 to_TO work_VVI out_RP some_DD of_IO the_AT others_NN2 ._. 
The_AT notation_NN1 is_VBZ fairly_RR standard_JJ but_CCB still_RR I_PPIS1 should_VM better_RRR say_VVI what_DDQ is_VBZ what_DDQ :_: H_ZZ1 ,_, magnetic_JJ field_NN1 strength_NN1 ;_; E_ZZ1 ,_, electric_JJ field_NN1 strength_NN1 ;_; D_ZZ1 ,_, electric_JJ flux_NN1 density_NN1 ;_; B_ZZ1 ,_, magnetic_JJ flux_NN1 density_NN1 ;_; J_ZZ1 ,_, current_JJ density_NN1 ;_; p_ZZ1 ,_, charge_NN1 density_NN1 ;_; F_ZZ1 ,_, force_NN1 ,_, q_ZZ1 ,_, charge_NN1 ;_; v_ZZ1 ,_, velocity_NN1 of_IO moving_VVG charge_NN1 ;_; u_ZZ1 ,_, permeability_NN1 ;_; e_ZZ1 ,_, permittivity_NN1 ._. 
The_AT two_MC latter_DA quantities_NN2 are_VBR constants_NN2 depending_II21 on_II22 the_AT material_NN1 under_II study_NN1 ._. 
Using_VVG the_AT subscript_NN1 zero_NN1 to_TO denote_VVI their_APPGE values_NN2 in_II free_JJ space_NN1 ,_, they_PPHS2 come_VV0 to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT rest_NN1 of_IO the_AT course_NN1 will_VM be_VBI concerned_JJ with_IW the_AT various_JJ solutions_NN2 of_IO eqns_NN2 (_( 1.1_MC )_) -(1.7)_NNU ._. 
Is_VBZ n't_XX this_DD1 boring_JJ for_IF an_AT1 engineer_NN1 ?_? 
Should_VM n't_XX this_DD1 be_VBI done_VDN by_II mathematicians_NN2 or_CC by_II computer_NN1 programmers_NN2 ?_? 
Not_XX for_RT41 the_RT42 time_RT43 being_RT44 ._. 
Perhaps_RR one_MC1 day_NNT1 computers_NN2 will_VM be_VBI big_JJ enough_RR and_CC numerical_JJ analysts_NN2 clever_JJ enough_RR so_CS21 that_CS22 the_AT engineer_NN1 will_VM only_RR have_VHI to_TO pose_VVI the_AT problem_NN1 ,_, but_CCB not_XX yet_RR ,_, and_CC not_XX ,_, I_PPIS1 think_VV0 ,_, for_IF some_DD time_NNT1 to_TO come_VVI ._. 
In_II the_AT large_JJ majority_NN1 of_IO cases_NN2 a_AT1 straightforward_JJ mathematical_JJ solution_NN1 is_VBZ just_RR out_II21 of_II22 the_AT question_NN1 ._. 
So_RR one_PN1 has_VHZ to_TO use_VVI that_DD1 delicate_JJ substance_NN1 known_VVN as_II physical_JJ intuition_NN1 ._. 
How_RRQ can_VM one_PN1 acquire_VVI physical_JJ intuition_NN1 ?_? 
There_EX is_VBZ no_AT easy_JJ way_NN1 ._. 
One_PN1 has_VHZ to_TO start_VVI with_IW a_AT1 simple_JJ physical_JJ configuration_NN1 ,_, solve_VV0 the_AT corresponding_JJ mathematical_JJ problem_NN1 ,_, then_RT solve_VV0 a_AT1 similar_JJ problem_NN1 and_CC then_RT another_DD1 problem_NN1 ,_, and_CC then_RT a_RR21 little_RR22 more_RGR difficult_JJ problem_NN1 ,_, and_CC soon_RR ._. 
The_AT first_MD breakthrough_NN1 comes_VVZ when_RRQ one_PN1 can_VM predict_VVI a_AT1 solution_NN1 without_IW actually_RR doing_VDG the_AT mathematics_NN1 ._. 
In_BCL21 order_BCL22 to_TO have_VHI a_AT1 unified_JJ view_NN1 of_IO the_AT subject_NN1 we_PPIS2 have_VH0 started_VVN with_IW Maxwell_NP1 's_GE equations_NN2 ._. 
It_PPH1 means_VVZ a_AT1 new_JJ approach_NN1 but_CCB not_XX a_AT1 radical_JJ departure_NN1 ._. 
The_AT subject_NN1 is_VBZ still_RR the_AT same_DA ._. 
You_PPY will_VM be_VBI able_JK to_TO see_VVI that_CST the_AT laws_NN2 you_PPY love_VV0 and_CC cherish_VV0 (_( Coulomb_NP1 's_GE ,_, Biot-Savart_NP1 's_GE ,_, Snell_NP1 's_GE ,_, etc._RA )_) all_DB follow_VV0 from_II our_APPGE eqns_NN2 (_( 1.1_MC )_) -(1.7)_NNU ._. 
The_AT order_NN1 of_IO discussion_NN1 will_VM follow_VVI the_AT traditional_JJ one_PN1 :_: electrostatics_NN2 first_MD ,_, followed_VVN by_II steady_JJ currents_NN2 ,_, then_RT we_PPIS2 shall_VM move_VVI on_RP to_II slowly_RR varying_JJ phenomena_NN2 ,_, and_CC reach_VV0 finally_RR the_AT most_RGT interesting_JJ part_NN1 ,_, fast-varying_JJ phenomena_NN2 ,_, exhibiting_VVG the_AT full_JJ beauty_NN1 of_IO Maxwell_NP1 's_GE wonderful_JJ equations._NNU 2_MC ._. 
Electrostatics_NN1 2.1_MC ._. 
Introduction_NN1 STATIC_JJ means_NN not_XX varying_VVG as_II a_AT1 function_NN1 of_IO time_NNT1 ._. 
So_RR all_DB our_APPGE quantities_NN2 p_ZZ1 ,_, J_ZZ1 ,_, E_ZZ1 ,_, B_ZZ1 ,_, D_ZZ1 ,_, and_CC H_ZZ1 will_VM be_VBI independent_JJ of_IO time_NNT1 ._. 
Is_VBZ there_EX such_DA a_AT1 thing_NN1 as_CSA time-independent_JJ charge_NN1 ?_? 
Yes_UH ,_, there_EX is_VBZ ._. 
It_PPH1 means_VVZ that_CST neither_RR the_AT magnitude_NN1 nor_CC the_AT position_NN1 of_IO the_AT charge_NN1 varies_VVZ as_II a_AT1 function_NN1 of_IO time_NNT1 ._. 
And_CC similarly_RR we_PPIS2 can_VM imagine_VVI constant_JJ electric_JJ and_CC magnetic_JJ fields_NN2 ._. 
Can_VM we_PPIS2 talk_VVI of_IO time-independent_JJ current_JJ ?_? 
Well_RR ,_, we_PPIS2 have_VH0 to_TO permit_VVI the_AT motion_NN1 of_IO charges_NN2 to_TO get_VVI any_DD current_JJ but_CCB if_CS the_AT amount_NN1 of_IO charge_NN1 crossing_VVG a_AT1 certain_JJ cross-section_NN1 is_VBZ always_RR the_AT same_DA then_RT the_AT current_JJ at_II that_DD1 point_NN1 is_VBZ independent_JJ of_IO time_NNT1 ._. 
On_II this_DD1 basis_NN1 constant_JJ currents_NN2 also_RR belong_VV0 to_II the_AT static_JJ branch_NN1 of_IO electricity_NN1 ._. 
It_PPH1 is_VBZ ,_, though_CS ,_, usual_JJ to_TO distinguish_VVI between_II electrostatics_NN2 and_CC magnetostatics_NN2 ;_; in_II the_AT former_DA case_NN1 the_AT variables_NN2 are_VBR p_ZZ1 ,_, E_ZZ1 ,_, and_CC D_ZZ1 ,_, whereas_CS in_II the_AT latter_DA case_NN1 they_PPHS2 are_VBR J_ZZ1 ,_, H_ZZ1 ,_, and_CC B._NP1 We_PPIS2 shall_VM now_RT proceed_VVI with_IW the_AT equations_NN2 of_IO electrostatics_NN2 ,_, which_DDQ may_VM be_VBI obtained_VVN from_II eqns_NN2 (_( 1.1_MC )_) -(1.7)_NNU by_II substituting_VVG a/at_FU =_FO 0_MC and_CC assuming_VVG that_CST v_ZZ1 ,_, J_ZZ1 ,_, H_ZZ1 ,_, and_CC B_ZZ1 are_VBR all_DB zero_MC ._. 
We_PPIS2 get_VV0 then_RT &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 shall_VM introduce_VVI now_RT a_AT1 scalar_JJ function_NN1 by_II the_AT relationship_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT physical_JJ significance_NN1 of_IO this_DD1 new_JJ function_NN1 may_VM be_VBI recognized_VVN by_II determining_VVG the_AT work_NN1 performed_VVN by_II carrying_VVG a_AT1 charge_NN1 from_II point_NN1 a_AT1 to_TO point_VVI b_ZZ1 :_: &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS the_AT negative_JJ sign_NN1 is_VBZ due_II21 to_II22 the_AT fact_NN1 that_CST the_AT work_NN1 is_VBZ done_VDN against_II the_AT electrical_JJ forces_NN2 ._. 
Substituting_VVG eqns_NN2 (_( 2.4_MC )_) and_CC (_( 2.5_MC )_) into_II eqn_NN1 (_( 2.6_MC )_) we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) are_VBR the_AT values_NN2 of_IO the_AT function_NN1 &lsqb;_( formula_NN1 &rsqb;_) at_II the_AT end-points_NN2 of_IO the_AT path_NN1 ._. 
We_PPIS2 have_VH0 used_VVN here_RL a_AT1 mathematical_JJ theorem_NN1 stating_VVG that_CST the_AT line_NN1 integral_JJ of_IO a_AT1 gradient_NN1 depends_VVZ only_RR on_II the_AT end-points_NN2 and_CC not_XX on_II the_AT connecting_JJ path_NN1 ._. 
The_AT potential_NN1 at_II point_NN1 b_ZZ1 may_VM be_VBI written_VVN with_IW the_AT aid_NN1 of_IO eqn_NN1 (_( 2.7_MC )_) in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Alternatively_RR ,_, we_PPIS2 may_VM choose_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, leading_VVG to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
What_DDQ is_VBZ the_AT good_JJ of_IO introducing_VVG &lsqb;_( formula_NN1 &rsqb;_) ?_? 
I._NP1 Owing_II21 to_II22 its_APPGE scalar_JJ character_NN1 it_PPH1 is_VBZ more_RGR easily_RR calculable_JJ than_CSN the_AT electric_JJ field_NN1 ._. 
Thus_RR very_RG often_RR in_II practice_NN1 we_PPIS2 determine_VV0 &lsqb;_( formula_NN1 &rsqb;_) first_MD and_CC E_ZZ1 afterwards_RT ,_, 2_MC ._. 
The_AT choice_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT form_NN1 of_IO eqn_NN1 (_( 2.5_MC )_) immediately_RR ensures_VVZ that_DD1 eqn_NN1 (_( 2.1_MC )_) is_VBZ satisfied_JJ :_: &lsqb;_( formula_NN1 &rsqb;_) ,_, so_CS we_PPIS2 have_VH0 less_DAR to_TO worry_VVI about_RP ._. 
If_CS the_AT potential_JJ function_NN1 is_VBZ so_RG useful_JJ should_VM n't_XX we_PPIS2 express_VVI all_DB our_APPGE equations_NN2 in_II31 terms_II32 of_II33 &lsqb;_( formula_NN1 &rsqb;_) ?_? 
Yes_UH ,_, in_II fact_NN1 this_DD1 is_VBZ what_DDQ other_JJ people_NN did_VDD in_II the_AT past_NN1 ._. 
So_RR let_VV0 us_PPIO2 convert_VVI eqn_NN1 (_( 2.2_MC )_) as_RR21 well_RR22 ._. 
Noting_VVG that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ is_VBZ known_VVN as_II Poisson_NP1 's_GE equation_NN1 ._. 
If_CS p=0_FO ,_, the_AT above_JJ equation_NN1 reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 equation_NN1 is_VBZ known_VVN after_II another_DD1 Frenchman_NN1 as_CSA Laplace_NP1 's_GE equation_NN1 ._. 
We_PPIS2 have_VH0 not_XX quite_RR finished_VVN ._. 
There_EX is_VBZ one_MC1 more_DAR equation_NN1 often_RR used_VVN in_II electrostatics_NN2 that_CST contains_VVZ charge_NN1 and_CC not_XX charge_NN1 density_NN1 ._. 
We_PPIS2 can_VM get_VVI it_PPH1 by_II integrating_VVG eqn_NN1 (_( 2.2_MC )_) over_II a_AT1 volume_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Using_VVG Gauss_NP1 '_GE theorem_NN1 for_IF the_AT left-hand_JJ side_NN1 and_CC noting_VVG that_CST the_AT volume_NN1 integral_JJ of_IO charge_NN1 density_NN1 is_VBZ just_RR the_AT total_JJ amount_NN1 of_IO charge_NN1 ,_, eqn_NN1 (_( 2.14_MC )_) takes_VVZ the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS S_ZZ1 is_VBZ the_AT closed_JJ surface_NN1 of_IO volume_NN1 and_CC q_ZZ1 is_VBZ the_AT charge_NN1 inside_RL ._. 
Eqn_NN1 (_( 2.15_MC )_) is_VBZ known_VVN as_II Gauss_NP1 's_GE law._NNU 2.2_MC ._. 
Coulomb_NP1 's_GE law_NN1 Having_VHG got_VVN the_AT equations_NN2 what_DDQ shall_VM we_PPIS2 do_VDI with_IW them_PPHO2 ?_? 
Let_VV0 us_PPIO2 first_MD try_VVI to_TO prove_VVI something_PN1 that_CST you_PPY have_VH0 come_VVN across_RL in_II school_NN1 ,_, Coulomb_NP1 's_GE law_NN1 ._. 
Note_VV0 the_AT difference_NN1 ._. 
We_PPIS2 shall_VM not_XX postulate_VVI Coulomb_NP1 's_GE law_NN1 ,_, we_PPIS2 are_VBR going_VVGK to_TO derive_VVI it_PPH1 ._. 
We_PPIS2 shall_VM have_VHI to_TO introduce_VVI point_NN1 charges_NN2 ,_, but_CCB let_VV0 us_PPIO2 be_VBI a_RR21 little_RR22 more_RGR general_JJ to_TO begin_VVI with_IW and_CC assume_VVI that_CST the_AT charge_NN1 is_VBZ uniformly_RR distributed_VVN within_II a_AT1 sphere_NN1 of_IO radius_NN1 r0_FO ._. 
We_PPIS2 shall_VM now_RT apply_VVI Gauss_NP1 '_GE law_NN1 and_CC take_VVI the_AT surface_NN1 of_IO integration_NN1 at_II a_AT1 radius_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Notice_VV0 that_CST everything_PN1 is_VBZ spherically_RR symmetric_JJ ,_, hence_RR D_ZZ1 must_VM be_VBI constant_JJ on_II the_AT chosen_JJ surface_NN1 ,_, and_CC the_AT integral_JJ comes_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS D=D_FO ._. 
In_II31 view_II32 of_II33 eqns_NN2 (_( 2.15_MC )_) and_CC (_( 2.16_MC )_) &lsqb;_( formula_NN1 &rsqb;_) or_CC ,_, in_II a_AT1 vacuum_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
Interestingly_RR ,_, the_AT electric_JJ field_NN1 does_VDZ not_XX depend_VVI on_II the_AT actual_JJ positions_NN2 of_IO the_AT charges_NN2 ._. 
As_CS31 long_CS32 as_CS33 the_AT charge_NN1 distribution_NN1 is_VBZ spherically_RR symmetric_JJ ,_, and_CC as_CS31 long_CS32 as_CS33 all_DB the_AT charges_NN2 are_VBR inside_II the_AT sphere_NN1 of_IO radius_NN1 r0_FO the_AT electric_JJ field_NN1 depends_VVZ only_RR on_II r_ZZ1 and_CC not_XX on_II r0_FO ._. 
So_RR we_PPIS2 can_VM just_RR as_RR21 well_RR22 imagine_VVI that_CST all_DB the_AT charge_NN1 is_VBZ concentrated_VVN at_II the_AT origin_NN1 of_IO the_AT coordinate_NN1 system_NN1 ._. 
Let_VV0 us_PPIO2 place_VVI now_RT another_DD1 bunch_NN1 of_IO charge_NN1 (_( say_VV0 q2_FO )_) into_II another_DD1 discrete_JJ point_NN1 a_AT1 distance_NN1 r12_FO away_II21 from_II22 our_APPGE first_MD charge_NN1 that_CST we_PPIS2 will_VM now_RT denote_VVI by_II q1_FO ._. 
With_IW the_AT aid_NN1 of_IO eqn_NN1 (_( 2.4_MC )_) we_PPIS2 get_VV0 for_IF the_AT force_NN1 upon_II &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC substituting_VVG for_IF E_ZZ1 from_II eqn_NN1 (_( 2.18_MC )_) we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
In_II words_NN2 :_: the_AT force_NN1 between_II two_MC charges_NN2 at_II rest_NN1 is_VBZ proportional_JJ to_II the_AT product_NN1 of_IO the_AT charges_NN2 and_CC inversely_RR proportional_JJ to_II the_AT square_NN1 of_IO the_AT distance_NN1 between_II them_PPHO2 ._. 
The_AT direction_NN1 of_IO the_AT force_NN1 is_VBZ obvious_JJ ,_, it_PPH1 can_VM only_RR act_VVI in_II the_AT line_NN1 connecting_VVG the_AT two_MC point_NN1 charges_NN2 ._. 
So_RR we_PPIS2 have_VH0 got_VVN Coulomb_NP1 's_GE law._NNU 2.3_MC ._. 
The_AT potential_NN1 due_II21 to_II22 charges_NN2 The_AT potential_NN1 due_II21 to_II22 a_AT1 point_NN1 charge_NN1 may_VM be_VBI determined_VVN with_IW the_AT aid_NN1 of_IO eqns_NN2 (_( 2.8_MC )_) and_CC (_( 2.18_MC )_) ,_, as_CSA follows_VVZ :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT usual_JJ convention_NN1 is_VBZ to_TO choose_VVI the_AT reference_NN1 point_NN1 at_II infinity_NN1 and_CC choose_VV0 the_AT corresponding_JJ potential_NN1 (_( a_ZZ1 )_) =_FO 0_MC ,_, so_CS we_PPIS2 get_VV0 for_IF the_AT potential_NN1 of_IO a_AT1 point_NN1 charge_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS we_PPIS2 have_VH0 a_AT1 number_NN1 of_IO point_NN1 charges_NN2 q1_FO ,_, q2_FO </c>_NULL ,_, qn_NNU at_II distances_NN2 r1_FO ,_, r2_FO </c>_NULL ,_, rn_NNU from_II the_AT point_NN1 where_RRQ we_PPIS2 wish_VV0 to_TO evaluate_VVI the_AT potential_NN1 then_RT we_PPIS2 can_VM simply_RR add_VVI all_DB the_AT potentials_NN2 ,_, leading_VVG to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS instead_II21 of_II22 point_NN1 charges_NN2 we_PPIS2 have_VH0 a_AT1 distributed_JJ space_NN1 charge_NN1 p_ZZ1 (_( x'_VV0 ,_, y'_PPY ,_, z'_NN1 )_) then_RT the_AT sum_NN1 in_II eqn_NN1 (_( 2.23_MC )_) goes_VVZ over_RP into_II an_AT1 integral_JJ :_: &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS r_ZZ1 is_VBZ now_RT the_AT distance_NN1 between_II the_AT elementary_JJ charge_NN1 &lsqb;_( formula_NN1 &rsqb;_) located_VVD at_II point_NN1 (_( x'_VV0 ,_, y'_PPY ,_, z'_NN1 )_) and_CC the_AT point_NN1 (_( x_ZZ1 ,_, y_ZZ1 ,_, z_ZZ1 )_) where_RRQ the_AT potential_NN1 is_VBZ evaluated_VVN ._. 
The_AT formulae_NN2 look_VV0 quite_RG reasonable_JJ ;_; what_DDQ you_PPY need_VV0 is_VBZ a_AT1 little_JJ practice_NN1 in_II handling_VVG them_PPHO2 ._. 
But_CCB as_CSA I_PPIS1 said_VVD before_II the_AT game_NN1 is_VBZ not_XX a_AT1 purely_RR mathematical_JJ one_PN1 ;_; it_PPH1 is_VBZ a_AT1 mixture_NN1 of_IO physics_NN1 and_CC mathematics_NN1 ,_, a_AT1 combination_NN1 of_IO intuition_NN1 and_CC technique_NN1 ._. 
So_RR before_CS we_PPIS2 embark_VV0 upon_II solving_VVG concrete_JJ examples_NN2 let_VV0 us_PPIO2 turn_VVI to_II a_AT1 graphical_JJ illustration_NN1 of_IO the_AT electric_JJ field_NN1 ._. 
We_PPIS2 shall_VM introduce_VVI field_NN1 lines_NN2 defined_VVN by_II the_AT statement_NN1 that_CST at_II each_DD1 point_NN1 on_II the_AT line_NN1 the_AT tangent_NN1 is_VBZ in_II the_AT direction_NN1 of_IO the_AT electric_JJ field_NN1 ._. 
The_AT magnitude_NN1 of_IO the_AT electric_JJ field_NN1 may_VM be_VBI represented_VVN at_II the_AT same_DA time_NNT1 by_II the_AT density_NN1 of_IO field_NN1 lines_NN2 ._. 
The_AT nearer_JJR they_PPHS2 are_VBR to_II each_PPX221 other_PPX222 the_AT greater_JJR is_VBZ the_AT field_NN1 ._. 
A_AT1 particularly_RR simple_JJ example_NN1 is_VBZ provided_VVN by_II a_AT1 point_NN1 charge_NN1 ._. 
The_AT electric_JJ field_NN1 is_VBZ always_RR in_II the_AT radial_JJ direction_NN1 ,_, so_CS the_AT field_NN1 lines_NN2 are_VBR just_RR straight_JJ lines_NN2 as_CSA shown_VVN in_II Fig._NN1 2.2_MC ._. 
According_II21 to_II22 convention_NN1 the_AT arrows_NN2 on_II the_AT lines_NN2 point_VV0 outwards_RL from_II a_AT1 positive_JJ point_NN1 charge_NN1 ._. 
Owing_II21 to_II22 radial_JJ symmetry_NN1 the_AT potential_NN1 is_VBZ constant_JJ on_II a_AT1 spherical_JJ surface_NN1 at_II a_AT1 distance_NN1 r_ZZ1 from_II the_AT point_NN1 charge_NN1 ._. 
Some_DD of_IO these_DD2 equipotential_JJ surfaces_NN2 are_VBR shown_VVN in_II Fig._NN1 2.2_MC with_IW dotted_JJ lines_NN2 ._. 
Notice_VV0 that_CST the_AT field_NN1 lines_NN2 are_VBR perpendicular_JJ to_II the_AT equipotential_JJ surfaces_NN2 ._. 
This_DD1 is_VBZ of_RR21 course_RR22 no_AT coincidence_NN1 ._. 
The_AT gradient_NN1 of_IO a_AT1 scalar_JJ function_NN1 is_VBZ a_AT1 vector_NN1 perpendicular_NN1 to_II the_AT &lsqb;_( formula_NN1 &rsqb;_) constant_JJ surface_NN1 and_CC points_NN2 in_II the_AT direction_NN1 of_IO increasing_JJ values_NN2 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
Since_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC in_II the_AT present_JJ case_NN1 q&gt;0_FO ,_, we_PPIS2 find_VV0 that_CST &lsqb;_( formula_NN1 &rsqb;_) decreases_VVZ with_IW increasing_JJ r_ZZ1 ._. 
Hence_RR the_AT vector_NN1 &lsqb;_( formula_NN1 &rsqb;_) points_VVZ inwards_RL and_CC &lsqb;_( formula_NN1 &rsqb;_) points_VVZ outwards_RL ._. 
So_RR the_AT whole_JJ picture_NN1 is_VBZ consistent_JJ ._. 
Having_VHG completed_VVN the_AT calculations_NN2 for_IF the_AT potential_NN1 of_IO a_AT1 point_NN1 charge_NN1 ,_, we_PPIS2 shall_VM now_RT investigate_VVI a_AT1 more_RGR complicated_JJ situation_NN1 ._. 
Yes_UH ,_, you_PPY guessed_VVD correctly_RR ,_, we_PPIS2 are_VBR going_VVGK to_TO investigate_VVI the_AT equipotential_JJ surfaces_NN2 and_CC field_NN1 lines_NN2 of_IO two_MC point_NN1 charges_NN2 ._. 
In_II31 view_II32 of_II33 eqn_NN1 (_( 2.23_MC )_) the_AT potential_NN1 due_II21 to_II22 two_MC point_NN1 charges_NN2 may_VM be_VBI written_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ for_IF the_AT coordinate_NN1 system_NN1 of_IO Fig._NN1 2.3_MC reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT equipotential_JJ surfaces_NN2 may_VM now_RT be_VBI plotted_VVN on_II the_AT basis_NN1 of_IO the_AT above_JJ equation_NN1 and_CC then_RT the_AT field_NN1 lines_NN2 may_VM be_VBI obtained_VVN as_CSA trajectories_NN2 orthogonal_JJ to_II the_AT equipotentials_NN2 ._. 
This_DD1 is_VBZ simple_JJ in_II principle_NN1 but_CCB extremely_RR messy_JJ and_CC tedious_JJ if_CS you_PPY want_VV0 to_TO solve_VVI it_PPH1 analytically_RR ._. 
The_AT best_JJT method_NN1 nowadays_RT ,_, of_RR21 course_RR22 ,_, is_VBZ to_TO put_VVI it_PPH1 on_II the_AT computer_NN1 and_CC let_VVI the_AT computer_NN1 plot_VVI the_AT lot_NN1 ._. 
For_IF q2=-q1_FO the_AT plot_NN1 in_II the_AT x_ZZ1 ,_, y_ZZ1 plane_NN1 is_VBZ shown_VVN in_II Fig._NN1 2.4_MC ._. 
If_CS you_PPY look_VV0 at_II the_AT Figure_NN1 and_CC study_VV0 it_PPH1 carefully_RR you_PPY will_VM be_VBI able_JK to_TO recognize_VVI a_AT1 number_NN1 of_IO &quot;_" commonsense_JJ &quot;_" points_NN2 ._. 
It_PPH1 is_VBZ common_JJ sense_NN1 once_CS you_PPY see_VV0 the_AT solution_NN1 but_CCB it_PPH1 is_VBZ unlikely_JJ that_CST you_PPY would_VM have_VHI thought_VVN of_IO all_DB of_IO them_PPHO2 a_JJ21 priori_JJ22 ._. 
For_REX21 example_REX22 ,_, 1._MC all_DB the_AT field_NN1 lines_NN2 originate_VV0 on_II the_AT positive_JJ charge_NN1 and_CC terminate_VV0 on_II the_AT negative_JJ charge_NN1 ;_; 2._MC the_AT potential_NN1 is_VBZ zero_MC on_II the_AT y=0_FO plane_NN1 ;_; 3._MC one_MC1 of_IO the_AT field_NN1 lines_NN2 coincides_VVZ with_IW part_NN1 of_IO the_AT y_ZZ1 axis_NN1 connecting_VVG the_AT two_MC charges._NNU 4_MC ._. 
The_AT x=0_FO and_CC y=0_FO planes_NN2 are_VBR planes_NN2 of_IO symmetry._NNU 5_MC ._. 
The_AT equipotential_JJ lines_NN2 become_VV0 more_DAR and_CC more_RGR similar_JJ to_II circles_NN2 as_CSA they_PPHS2 get_VV0 nearer_II21 to_II22 either_DD1 charge._NNU 6_MC ._. 
Just_RR behind_II the_AT charges_NN2 the_AT effect_NN1 of_IO the_AT opposite_JJ charge_NN1 is_VBZ minimal_JJ so_CS the_AT field_NN1 lines_NN2 resemble_VV0 those_DD2 of_IO an_AT1 isolated_JJ charge_NN1 ._. 
The_AT better_NN1 you_PPY grasp_VV0 the_AT salient_JJ points_NN2 (_( and_CC store_VV0 them_PPHO2 in_II your_APPGE memory_NN1 )_) ,_, the_AT more_RGR physical_JJ intuition_NN1 and_CC predictive_JJ power_NN1 you_PPY will_VM acquire_VVI ._. 
Should_VM n't_XX one_PN1 rely_VVI solely_RR on_II mathematics_NN1 ?_? 
Physical_JJ intuition_NN1 might_VM provide_VVI part_NN1 of_IO the_AT answer_NN1 ,_, but_CCB surely_RR mathematics_NN1 will_VM always_RR give_VVI the_AT complete_JJ answer_NN1 ._. 
This_DD1 is_VBZ true_JJ for_IF simple_JJ problems_NN2 but_CCB ,_, as_CSA I_PPIS1 have_VH0 said_VVN many_DA2 times_NNT2 before_RT ,_, as_CS31 soon_CS32 as_CS33 the_AT problems_NN2 become_VV0 more_RRR complicated_VVD our_APPGE mathematical_JJ knowledge_NN1 turns_VVZ out_RP to_TO be_VBI greatly_RR deficient_JJ ._. 
In_II practice_NN1 you_PPY will_VM find_VVI that_CST problems_NN2 may_VM be_VBI broadly_RR divided_VVN into_II two_MC classes_NN2 :_: (_( I_ZZ1 )_) problems_NN2 that_CST have_VH0 been_VBN solved_VVN ,_, and_CC (_( II_MC )_) problems_NN2 that_CST are_VBR insoluble_JJ ._. 
The_AT reasons_NN2 for_IF belonging_VVG into_II class_NN1 II_MC may_VM be_VBI numerous_JJ :_: you_PPY can_VM not_XX formulate_VVI the_AT problem_NN1 ,_, you_PPY can_VM not_XX solve_VVI the_AT resulting_JJ equations_NN2 analytically_RR ,_, the_AT computer_NN1 you_PPY have_VH0 access_NN1 to_TO is_VBZ not_XX big_JJ enough_RR ,_, etc._RA ,_, etc_RA ._. 
It_PPH1 all_DB boils_VVZ down_RP to_II the_AT fact_NN1 that_CST the_AT problem_NN1 needs_VVZ to_TO be_VBI simplified_VVN ._. 
You_PPY will_VM find_VVI that_CST you_PPY never_RR solve_VV0 the_AT original_JJ problem_NN1 ._. 
At_RR21 best_RR22 you_PPY solve_VV0 a_AT1 similar_JJ one_PN1 ._. 
But_CCB how_RRQ can_VM you_PPY recognize_VVI a_AT1 &quot;_" similar_JJ &quot;_" problem_NN1 ?_? 
Which_DDQ simplifications_NN2 are_VBR admissible_JJ without_IW losing_VVG the_AT essential_JJ characteristics_NN2 of_IO the_AT problem_NN1 ?_? 
For_IF all_DB that_CST you_PPY need_VV0 intuition_NN1 ._. 
The_AT electric_JJ dipole_NN1 One_PN1 often_RR goes_VVZ to_II extremes_NN2 in_BCL21 order_BCL22 to_TO arrive_VVI at_II a_AT1 physical_JJ configuration_NN1 that_CST is_VBZ mathematically_RR soluble_JJ by_II simple_JJ means_NN ._. 
This_DD1 is_VBZ what_DDQ we_PPIS2 are_VBR going_VVGK to_TO do_VDI now_RT and_CC besides_RR assuming_VVG that_CST q=q1=-q2_FO we_PPIS2 will_VM also_RR state_VVI that_CST the_AT two_MC opposite_JJ charges_NN2 are_VBR infinitesimally_RR close_JJ to_II each_PPX221 other_PPX222 ._. 
Is_VBZ such_DA a_AT1 situation_NN1 entirely_RR fictitious_JJ ?_? 
No_UH ,_, it_PPH1 can_VM occur_VVI in_II practice_NN1 ._. 
But_CCB surely_RR ,_, two_MC charges_NN2 will_VM never_RR be_VBI infinitesimally_RR close_JJ to_II each_PPX221 other_PPX222 ._. 
Of_RR21 course_RR22 not_XX ,_, all_DB I_PPIS1 mean_VV0 is_VBZ that_CST the_AT distance_NN1 between_II the_AT charges_NN2 is_VBZ very_RG small_JJ with_II31 respect_II32 to_II33 some_DD other_JJ distance_NN1 of_IO interest_NN1 ._. 
For_REX21 example_REX22 ,_, the_AT two_MC charges_NN2 may_VM be_VBI of_IO some_DD atomic_JJ distance_NN1 (_( of_IO the_AT order_NN1 of_IO 10_MC -10_MC m_NNU )_) apart_RL ,_, whereas_CS we_PPIS2 are_VBR interested_JJ in_II their_APPGE effect_NN1 at_II macroscopic_JJ distances_NN2 ;_; or_CC take_VV0 the_AT so-called_JJ dipole_NN1 aerial_NN1 where_CS the_AT assumed_JJ separation_NN1 of_IO charges_NN2 is_VBZ small_JJ in_II31 comparison_II32 with_II33 the_AT wavelength_NN1 of_IO oscillation_NN1 ._. 
We_PPIS2 shall_VM return_VVI to_II the_AT latter_DA ,_, time-varying_JJ problem_NN1 in_II Section_NN1 5.16_MC ;_; let_VV0 us_PPIO2 first_MD solve_VVI here_RL the_AT static_JJ case_NN1 ._. 
If_CS we_PPIS2 are_VBR considering_VVG distances_NN2 far_RR away_II21 from_II22 the_AT origin_NN1 of_IO Fig._NN1 2.3_MC ,_, i.e._REX &lsqb;_( formula_NN1 &rsqb;_) ,_, using_VVG the_AT approximation_NN1 &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 obtain_VV0 with_IW which_DDQ eqn_NN1 (_( 2.26_MC )_) reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT product_NN1 qd_NNU is_VBZ called_VVN the_AT electric_JJ dipole_NN1 moment_NN1 ,_, and_CC we_PPIS2 shall_VM denote_VVI it_PPH1 by_II pe_NN1 ._. 
We_PPIS2 shall_VM also_RR introduce_VVI a_AT1 vector_NN1 pe=qd_FO ,_, where_CS d_ZZ1 is_VBZ the_AT vector_NN1 connecting_VVG the_AT two_MC point_NN1 charges_NN2 located_VVN at_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
By_II definition_NN1 the_AT vector_NN1 points_VVZ from_II the_AT negative_JJ to_II the_AT positive_JJ charge_NN1 ._. 
With_IW the_AT aid_NN1 of_IO this_DD1 vector_NN1 we_PPIS2 can_VM replace_VVI &lsqb;_( formula_NN1 &rsqb;_) by_II &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT unit_NN1 vector_NN1 in_II the_AT direction_NN1 of_IO point_NN1 P_ZZ1 at_II a_AT1 radius_NN1 r_ZZ1 ._. 
So_RR we_PPIS2 may_VM write_VVI eqn_NN1 (_( 2.30_MC )_) in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Yet_RR another_DD1 alternative_JJ form_NN1 is_VBZ obtained_VVN by_II using_VVG spherical_JJ coordinates_NN2 &lsqb;_( formula_NN1 &rsqb;_) in_II which_DDQ &lsqb;_( formula_NN1 &rsqb;_) appears_VVZ as_CSA &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT above_JJ form_NN1 is_VBZ probably_RR best_RRT suited_VVN for_IF deriving_VVG the_AT components_NN2 of_IO the_AT electric_JJ field_NN1 ._. 
Using_VVG the_AT formulae_NN2 (_( A_ZZ1 11_MC and_CC 14_MC )_) for_IF the_AT gradient_NN1 in_II spherical_JJ coordinates_NN2 ,_, we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
&quot;_" The_AT potential_JJ far_JJ away_II21 from_II22 the_AT charges_NN2 If_CS the_AT charges_NN2 are_VBR within_II a_AT1 finite_JJ volume_NN1 and_CC we_PPIS2 wish_VV0 to_TO determine_VVI the_AT potential_NN1 at_II a_AT1 point_NN1 P_ZZ1 ,_, far_RR away_II21 from_II22 this_DD1 volume_NN1 ,_, the_AT formula_NN1 we_PPIS2 have_VH0 obtained_VVN before_II (_( eqn_NN1 (_( 2.23_MC )_) )_) may_VM be_VBI simplified_VVN ._. 
We_PPIS2 shall_VM assume_VVI (_( Fig._NN1 2.6_MC )_) that_CST the_AT vectors_NN2 drawn_VVN from_II any_DD point_NN1 charge_NN1 to_TO point_VVI P_ZZ1 are_VBR all_DB parallel_RR to_II the_AT vector_NN1 r0_FO drawn_VVN from_II an_AT1 arbitrarily_RR chosen_VVN origin_NN1 inside_RL (_( just_RR another_DD1 way_NN1 of_IO saying_VVG that_CST the_AT point_NN1 P_ZZ1 is_VBZ far_RR away_RL )_) ._. 
This_DD1 is_VBZ a_AT1 technique_NN1 often_RR used_VVN that_CST leads_VVZ to_II a_AT1 quick_JJ and_CC simple_JJ answer_NN1 ._. 
We_PPIS2 may_VM then_RT express_VVI the_AT individual_JJ distances_NN2 as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ di_FW is_VBZ the_AT vector_NN1 giving_VVG the_AT position_NN1 of_IO charge_NN1 qi_NN2 ._. 
With_IW the_AT aid_NN1 of_IO the_AT above_JJ relation_NN1 we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ substituted_VVD into_II eqn_NN1 (_( 2.23_MC )_) yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Thus_RR we_PPIS2 have_VH0 a_AT1 contribution_NN1 depending_II21 on_II22 the_AT total_JJ charge_NN1 ,_, and_CC a_AT1 second_MD term_NN1 bearing_VVG strong_JJ resemblance_NN1 to_II our_APPGE dipole_NN1 formulae_NN2 ._. 
Since_CS the_AT second_MD term_NN1 decays_VVZ as_II 1/r2_FU ,_, could_VM it_PPH1 ever_RR become_VVI important_JJ ?_? 
Yes_UH ,_, for_IF many_DA2 charge_NN1 distributions_NN2 of_IO practical_JJ interest_NN1 the_AT net_JJ charge_NN1 is_VBZ zero_MC ,_, so_CS the_AT first_MD term_NN1 disappears_VVZ and_CC the_AT second_MD term_NN1 acquires_VVZ significance_NN1 ._. 
For_IF two_MC charges_NN2 of_IO opposite_JJ sign_NN1 eqn_NN1 (_( 2.36_MC )_) reduces_VVZ to_II eqn_NN1 (_( 2.31_MC )_) ;_; we_PPIS2 just_RR have_VH0 the_AT dipole_NN1 potential_NN1 previously_RR derived_VVN ._. 
For_IF a_AT1 large_JJ number_NN1 of_IO charges_NN2 we_PPIS2 sum_VV0 up_RP the_AT contribution_NN1 of_IO each_DD1 dipole_NN1 moment_NN1 ._. 
Multipoles_NN2 Why_RRQ stop_VVI at_II dipoles_NN2 ?_? 
Could_VM we_PPIS2 have_VHI higher-order_JJ moments_NN2 as_RR21 well_RR22 ?_? 
The_AT answer_NN1 is_VBZ yes_UH ,_, but_CCB the_AT mathematics_NN1 gets_VVZ more_DAR and_CC more_RGR tedious_JJ ._. 
A_AT1 not-too-difficult_JJ example_NN1 is_VBZ a_AT1 special_JJ sort_NN1 of_IO quadrupole_NN1 (_( two_MC dipoles_NN2 of_IO equal_JJ dipole_NN1 moment_NN1 and_CC of_IO opposite_JJ directions_NN2 arranged_VVD axially_RR )_) shown_VVN in_II Fig._NN1 2.7_MC ._. 
Then_RT eqn_NN1 (_( 2.36_MC )_) would_VM give_VVI zero_NN1 and_CC we_PPIS2 need_VV0 better_JJR approximations_NN2 for_IF ri_NN2 ._. 
If_CS you_PPY are_VBR good_JJ at_II expanding_JJ functions_NN2 up_RG21 to_RG22 second_MD order_NN1 you_PPY might_VM like_VVI to_TO attempt_VVI Example_NN1 2.5._MC 2.4_MC The_AT electric_JJ field_NN1 due_II21 to_II22 a_AT1 line_NN1 charge_NN1 Let_VV0 us_PPIO2 take_VVI as_II an_AT1 example_NN1 an_AT1 infinitely_RR long_RR ,_, infinitely_RR thin_JJ distribution_NN1 of_IO charges_NN2 as_CSA shown_VVN in_II Fig._NN1 2.8_MC ._. 
Let_VV0 us_PPIO2 further_RRR assume_VVI that_CST the_AT charge_NN1 distribution_NN1 is_VBZ uniform_JJ and_CC denote_VV0 the_AT charge_NN1 per_II unit_NN1 length_NN1 by_II pl_NN1 (_( we_PPIS2 are_VBR defining_VVG thereby_RR a_AT1 linear_JJ charge_NN1 density_NN1 of_IO dimension_NN1 coulomb_NN1 per_II metre_NNU1 )_) ._. 
Our_APPGE aim_NN1 is_VBZ to_TO determine_VVI the_AT electric_JJ field_NN1 ._. 
One_MC1 possible_JJ method_NN1 is_VBZ to_TO work_VVI out_RP the_AT electric_JJ field_NN1 due_II21 to_II22 a_AT1 point_NN1 charge_NN1 pi_NN1 dz_NNU located_VVN at_II z_ZZ1 and_CC then_RT add_VV0 the_AT field_NN1 due_II21 to_II22 all_DB the_AT other_JJ point_NN1 charges_NN2 present_NN1 ._. 
We_PPIS2 obtain_VV0 then_RT for_IF the_AT electric_JJ field_NN1 at_II the_AT coordinate_NN1 R_ZZ1 ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT unit_NN1 vector_NN1 in_II the_AT direction_NN1 of_IO the_AT r_ZZ1 vector_NN1 ._. 
Owing_II21 to_II22 symmetry_NN1 considerations_NN2 (_( there_EX is_VBZ the_AT same_DA infinite_JJ amount_NN1 of_IO charge_NN1 in_II the_AT region_NN1 &lsqb;_( formula_NN1 &rsqb;_) as_CSA in_II the_AT corresponding_JJ region_NN1 &lsqb;_( formula_NN1 &rsqb;_) )_) the_AT electric_JJ field_NN1 can_VM have_VHI no_AT component_NN1 in_II the_AT z_ZZ1 direction_NN1 ,_, so_CS we_PPIS2 need_VV0 only_RR to_II worry_NN1 about_II the_AT radial_JJ component_NN1 ._. 
This_DD1 we_PPIS2 may_VM obtain_VVI easily_RR from_II Fig._NN1 2.8_MC as_CSA follows_VVZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 is_VBZ preferable_JJ to_TO do_VDI the_AT integration_NN1 for_IF so_RR we_PPIS2 shall_VM rewrite_VVI eqn_NN1 (_( 2.38_MC )_) with_IW the_AT aid_NN1 of_IO the_AT relation_NN1 &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ may_VM be_VBI integrated_VVN between_II the_AT limits_NN2 =-_FO ;_; /2_MF ;_; and_CC =_FO ;_; /2_MF ;_; to_TO yield_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 formula_NN1 tells_VVZ us_PPIO2 a_AT1 lot_NN1 and_CC in_II simple_JJ language_NN1 too_RR ._. 
The_AT only_JJ surviving_JJ component_NN1 of_IO the_AT electric_JJ field_NN1 varies_VVZ inversely_RR with_IW the_AT distance_NN1 from_II the_AT line_NN1 charge_NN1 and_CC it_PPH1 is_VBZ independent_JJ of_IO the_AT coordinate_NN1 ._. 
Remember_VV0 ,_, for_IF a_AT1 point_NN1 charge_NN1 the_AT electric_JJ field_NN1 varies_VVZ with_IW the_AT inverse_JJ square_NN1 of_IO the_AT distance_NN1 ,_, but_CCB it_PPH1 is_VBZ just_RR inverse_JJ distance_NN1 for_IF the_AT line_NN1 charge_NN1 ._. 
Why_RRQ is_VBZ this_DD1 worth_II remembering_VVG ?_? 
Because_CS the_AT relationships_NN2 are_VBR simple_JJ ,_, they_PPHS2 will_VM not_XX considerably_RR burden_VVI your_APPGE memory_NN1 ,_, and_CC at_II the_AT same_DA time_NNT1 will_VM assist_VVI you_PPY in_II building_VVG up_RP your_APPGE intuitive_JJ picture_NN1 ._. 
Take_VV0 now_RT a_AT1 finite_JJ line_NN1 charge_NN1 as_CSA shown_VVN in_II Fig._NN1 2.9(a)_FO and_CC (_( b_ZZ1 )_) using_VVG different_JJ scales_NN2 ._. 
Can_VM you_PPY answer_VVI the_AT question_NN1 what_DDQ is_VBZ the_AT relative_JJ strength_NN1 of_IO the_AT electric_JJ field_NN1 at_II points_NN2 N_ZZ1 and_CC M_ZZ1 in_II Fig._NN1 2.9(a)_FO and_CC at_II points_NN2 Q_ZZ1 and_CC P_ZZ1 in_II Fig._NN1 2.9(b)_FO ?_? 
To_TO be_VBI concrete_JJ let_VV0 us_PPIO2 take_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 looks_VVZ from_II points_NN2 N_ZZ1 and_CC M_ZZ1 as_CS21 if_CS22 the_AT linear_JJ charge_NN1 distribution_NN1 would_VM be_VBI very_RG long_RR indeed_RR ,_, perhaps_RR infinitely_RR long_JJ ._. 
Thus_RR the_AT electric_JJ field_NN1 distribution_NN1 may_VM be_VBI expected_VVN to_TO follow_VVI a_AT1 (_( distance_NN1 )_) -1_MC law_NN1 ._. 
Hence_RR En/Em=_FU ._. 
Looking_VVG from_II points_NN2 Q_ZZ1 and_CC P_ZZ1 the_AT linear_JJ charge_NN1 distribution_NN1 between_II z=-H_FO and_CC z=H_FO appears_VVZ more_RRR like_II a_AT1 point_NN1 charge_NN1 concentrated_VVN at_II 0_MC ._. 
Thus_RR the_AT electric_JJ field_NN1 distribution_NN1 may_VM be_VBI expected_VVN to_TO follow_VVI a_AT1 (_( distance_NN1 )_) -2_MC law_NN1 ._. 
Hence_RR Eq/Ep=_NN1 ._. 
We_PPIS2 have_VH0 now_RT been_VBN able_JK to_TO give_VVI immediate_JJ answers_NN2 to_II fairly_RR complicated_JJ questions_NN2 ._. 
How_RGQ good_JJ are_VBR the_AT approximations_NN2 ?_? 
Derive_VV0 the_AT exact_JJ formula_NN1 for_IF the_AT electric_JJ field_NN1 ,_, work_VV0 out_RP its_APPGE values_NN2 at_II points_NN2 M_ZZ1 ,_, N_ZZ1 ,_, P_ZZ1 ,_, Q_ZZ1 ,_, take_VV0 the_AT ratios_NN2 ,_, and_CC see_VV0 for_IF yourself_PPX1 how_RRQ good_JJ our_APPGE approximations_NN2 are_VBR ._. 
Let_VV0 us_PPIO2 solve_VVI now_RT the_AT same_DA problem_NN1 by_II another_DD1 method_NN1 ._. 
Instead_II21 of_II22 calculating_VVG the_AT electric_JJ field_NN1 directly_RR from_II the_AT charge_NN1 distribution_NN1 let_VV0 us_PPIO2 first_MD determine_VVI the_AT potential_NN1 and_CC obtain_VV0 the_AT electric_JJ field_NN1 by_II differentiation_NN1 ._. 
This_DD1 is_VBZ surely_RR the_AT better_JJR method_NN1 because_CS we_PPIS2 wo_VM n't_XX worry_VVI about_II vectors_NN2 when_CS doing_VDG the_AT integration_NN1 ._. 
In_BCL21 order_BCL22 to_TO avoid_VVI a_AT1 lot_NN1 of_IO painful_JJ algebra_NN1 we_PPIS2 shall_VM further_RRR simplify_VVI the_AT problem_NN1 and_CC work_VVI out_RP the_AT potential_NN1 for_IF &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT potential_NN1 of_IO a_AT1 point_NN1 charge_NN1 is_VBZ given_VVN by_II eqn_NN1 (_( 2.22_MC )_) ,_, and_CC that_DD1 of_IO a_AT1 charge_NN1 distribution_NN1 by_II eqn_NN1 (_( 2.24_MC )_) ._. 
Whichever_DDQV we_PPIS2 use_VV0 we_PPIS2 end_VV0 up_RP with_IW the_AT following_JJ integration_NN1 for_IF the_AT potential_NN1 of_IO an_AT1 infinite_JJ line_NN1 charge_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
All_DB we_PPIS2 need_VV0 to_TO do_VDI now_RT is_VBZ to_TO put_VVI in_II the_AT limits_NN2 ,_, but_CCB alas_UH &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 get_VV0 the_AT result_NN1 that_CST the_AT potential_NN1 at_II point_NN1 P_ZZ1 is_VBZ infinitely_RR large_JJ ._. 
Have_VH0 we_PPIS2 done_VDN something_PN1 wrong_JJ ?_? 
No_UH ,_, we_PPIS2 have_VH0 done_VDN the_AT same_DA thing_NN1 as_CSA before_RT with_IW the_AT only_JJ difference_NN1 that_CST we_PPIS2 calculated_VVD the_AT potential_NN1 instead_II21 of_II22 the_AT electric_JJ field_NN1 ._. 
Where_RRQ could_VM the_AT trouble_NN1 lie_VVI ?_? 
Surely_RR ,_, in_II the_AT infinite_JJ nature_NN1 of_IO our_APPGE line_NN1 source_NN1 ._. 
There_EX are_VBR no_AT infinitely_RR long_JJ line_NN1 charges_NN2 in_II nature_NN1 ._. 
We_PPIS2 have_VH0 taken_VVN an_AT1 unphysical_JJ picture_NN1 and_CC we_PPIS2 get_VV0 a_AT1 nonsensical_JJ answer_NN1 ._. 
But_CCB why_RRQ did_VDD we_PPIS2 get_VVI a_AT1 reasonable_JJ result_NN1 for_IF the_AT electric_JJ field_NN1 ?_? 
That_DD1 was_VBDZ calculated_VVN for_IF an_AT1 infinitely_RR long_JJ line_NN1 charge_NN1 too_RR ._. 
Well_RR ,_, sometimes_RT you_PPY get_VV0 away_RL with_IW it_PPH1 ,_, sometimes_RT you_PPY do_VD0 n't_XX ._. 
Why_RRQ ?_? 
The_AT cause_NN1 is_VBZ only_RR known_VVN to_II mathematicians_NN2 and_CC philosophers_NN2 constantly_RR engaged_VVN in_II the_AT study_NN1 of_IO infinity_NN1 ._. 
What_DDQ can_VM an_AT1 ordinary_JJ physicist_NN1 or_CC engineer_NN1 do_VD0 ?_? 
Well_RR ,_, there_EX are_VBR several_DA2 avenues_NN2 open_VV0 ._. 
Number_NN1 one_MC1 is_VBZ to_TO acknowledge_VVI the_AT fact_NN1 that_CST our_APPGE line_NN1 charge_NN1 is_VBZ not_XX infinitely_RR long_JJ ,_, integrate_VV0 between_II the_AT limits_NN2 -H_NP1 and_CC +H_FO ,_, differentiate_VV0 to_TO obtain_VVI the_AT electric_JJ field_NN1 ,_, and_CC let_VVI then_RT H_ZZ1 go_VV0 to_II infinity_NN1 ._. 
Then_RT if_CS there_EX is_VBZ any_DD justice_NN1 on_II earth_NN1 ,_, we_PPIS2 shall_VM arrive_VVI at_II eqn_NN1 (_( 2.41_MC )_) ._. 
The_AT other_JJ thing_NN1 we_PPIS2 can_VM do_VDI is_VBZ to_TO retrace_VVI our_APPGE steps_NN2 leading_VVG to_II eqn_NN1 (_( 2.22_MC )_) ._. 
We_PPIS2 need_VV0 to_TO go_VVI only_RR as_CS31 far_CS32 as_CS33 eqn_NN1 (_( 2.21_MC )_) ._. 
We_PPIS2 may_VM see_VVI then_RT that_CST we_PPIS2 chose_VVD our_APPGE reference_NN1 point_NN1 at_II infinity_NN1 ._. 
Perhaps_RR there_EX is_VBZ the_AT rub_NN1 ._. 
We_PPIS2 may_VM have_VHI tampered_VVN too_RG much_DA1 with_IW infinity_NN1 ._. 
Let_VV0 us_PPIO2 abandon_VVI that_DD1 assumption_NN1 and_CC see_VVI what_DDQ happens_VVZ ._. 
Thus_RR we_PPIS2 are_VBR going_VVGK to_TO say_VVI that_CST (_( a_ZZ1 )_) =_FO 0_MC at_II some_DD other_JJ point_NN1 ._. 
It_PPH1 probably_RR matters_VVZ little_RR where_CS we_PPIS2 choose_VV0 that_DD1 point_NN1 as_CS31 long_CS32 as_CS33 it_PPH1 is_VBZ not_XX at_II infinity_NN1 ._. 
So_RR let_VV0 us_PPIO2 choose_VVI it_PPH1 for_IF convenience_NN1 at_II z_ZZ1 =_FO 0_MC at_II a_AT1 distance_NN1 R0_FO from_II the_AT line_NN1 charge_NN1 as_CSA shown_VVN in_II Fig._NN1 2.10_MC ._. 
Then_RT the_AT potential_NN1 at_II P_ZZ1 due_II21 to_II22 a_AT1 point_NN1 charge_NN1 at_II z_ZZ1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC the_AT potential_NN1 due_II21 to_II22 the_AT infinite_JJ line_NN1 charge_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Our_APPGE attempts_NN2 at_II determining_VVG both_DB2 the_AT electric_JJ field_NN1 strength_NN1 and_CC the_AT potential_NN1 have_VH0 started_VVN with_IW considering_VVG the_AT effect_NN1 of_IO a_AT1 single_JJ point_NN1 charge_NN1 and_CC have_VH0 been_VBN followed_VVN by_II an_AT1 integration_NN1 for_IF obtaining_VVG the_AT total_JJ effect_NN1 of_IO the_AT infinitely_RR long_JJ line_NN1 charge_NN1 ._. 
Is_VBZ there_RL another_DD1 ,_, more_RGR direct_JJ way_NN1 of_IO determining_VVG the_AT electric_JJ field_NN1 ?_? 
Well_RR ,_, there_EX is_VBZ Gauss_NP1 's_GE law_NN1 ,_, we_PPIS2 have_VH0 n't_XX tried_VVN using_VVG that_DD1 ._. 
If_CS the_AT line_NN1 charge_NN1 is_VBZ infinitely_RR long_JJ (_( so_CS21 that_CS22 the_AT field_NN1 depends_VVZ on_II the_AT radius_NN1 only_RR )_) we_PPIS2 can_VM choose_VVI our_APPGE Gaussian_JJ surface_NN1 as_II a_AT1 cylinder_NN1 wrapped_VVN round_II the_AT line_NN1 charge_NN1 (_( Fig._NN1 2.11_MC )_) ._. 
By_II relying_VVG on_II circular_JJ symmetry_NN1 we_PPIS2 can_VM further_RRR claim_VVI that_CST the_AT electric_JJ field_NN1 will_VM have_VHI only_RR a_AT1 radial_JJ component_NN1 which_DDQ means_VVZ that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT azimuth_NN1 angle_NN1 in_II the_AT cylindrical_JJ coordinate_NN1 system_NN1 ._. 
The_AT application_NN1 of_IO Gauss_NP1 's_GE law_NN1 (_( eqn_NN1 (_( 2.15_MC )_) )_) leads_VVZ then_RT to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Performing_VVG the_AT integration_NN1 in_II the_AT z_ZZ1 direction_NN1 for_IF any_DD finite_JJ length_NN1 ,_, we_PPIS2 get_VV0 for_IF the_AT electric_JJ field_NN1 strength_NN1 &lsqb;_( formula_NN1 &rsqb;_) in_II agreement_NN1 with_IW eqn_NN1 (_( 2.41_MC )_) ._. 
The_AT potential_NN1 may_VM be_VBI obtained_VVN from_II the_AT electric_JJ field_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 appears_VVZ that_CST the_AT application_NN1 of_IO Gauss_NP1 's_GE law_NN1 leads_VVZ much_RR more_RGR quickly_RR to_II the_AT required_JJ result_NN1 ._. 
What_DDQ is_VBZ the_AT moral_NN1 of_IO the_AT story_NN1 ?_? (_( i_ZZ1 )_) Tamper_VV0 with_IW infinity_NN1 at_II your_APPGE own_DA peril_NN1 ,_, and_CC (_( ii_MC )_) Some_DD ways_NN2 of_IO solving_VVG a_AT1 problem_NN1 are_VBR easier_RRR than_CSN others._NNU 2.5_MC ._. 
The_AT electric_JJ field_NN1 due_II21 to_II22 a_AT1 sheet_NN1 of_IO charge_NN1 We_PPIS2 shall_VM now_RT investigate_VVI the_AT case_NN1 when_CS the_AT charge_NN1 is_VBZ uniformly_RR distributed_VVN over_II a_AT1 plane_NN1 (_( the_AT x_ZZ1 ,_, y_ZZ1 plane_NN1 in_II Fig._NN1 2.12_MC )_) ._. 
The_AT charge_NN1 extends_VVZ to_II an_AT1 infinitesimal_JJ distance_NN1 in_II the_AT z_ZZ1 direction_NN1 so_CS we_PPIS2 shall_VM call_VVI it_PPH1 a_AT1 surface_NN1 charge_NN1 ._. 
To_TO be_VBI consistent_JJ with_IW our_APPGE previous_JJ notation_NN1 for_IF a_AT1 line_NN1 charge_NN1 we_PPIS2 shall_VM denote_VVI it_PPH1 by_II ps_NNU2 (_( dimension_NN1 coulomb_NN1 m-2_FO )_) ._. 
The_AT only_JJ component_NN1 of_IO the_AT electric_JJ field_NN1 is_VBZ in_II the_AT z_ZZ1 direction_NN1 (_( the_AT others_NN2 must_VM be_VBI zero_NN1 by_II symmetry_NN1 considerations_NN2 )_) ._. 
If_CS the_AT sheet_NN1 consists_VVZ of_IO positive_JJ charge_NN1 the_AT electric_JJ field_NN1 points_VVZ outwards_RL ._. 
Since_CS the_AT space_NN1 to_II the_AT right_NN1 of_IO the_AT charged_JJ sheet_NN1 looks_VVZ the_AT same_DA as_CSA that_DD1 to_II the_AT left_JJ ,_, the_AT magnitude_NN1 of_IO the_AT electric_JJ field_NN1 will_VM be_VBI the_AT same_DA at_II z=a_FO and_CC z=-a_FO ._. 
Learning_VVG from_II our_APPGE meandering_NN1 in_II the_AT previous_JJ section_NN1 we_PPIS2 shall_VM start_VVI straight_RR21 away_RR22 with_IW Gauss_NP1 's_GE law_NN1 ._. 
Choosing_VVG our_APPGE surface_NN1 as_II the_AT box_NN1 of_IO Fig.2.13_FO we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
There_EX is_VBZ no_AT contribution_NN1 from_II the_AT side_NN1 surfaces_NN2 because_CS the_AT scalar_JJ product_NN1 E.dS_NN2 vanishes_VVZ ._. 
The_AT total_JJ charge_NN1 enclosed_VVN is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, hence_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT electric_JJ field_NN1 is_VBZ a_AT1 constant_JJ everywhere_RL in_II space_NN1 but_CCB is_VBZ of_IO different_JJ sign_NN1 on_II the_AT two_MC sides_NN2 of_IO the_AT sheet_NN1 ._. 
Let_VV0 us_PPIO2 put_VVI now_RT a_AT1 sheet_NN1 of_IO opposite_JJ charge_NN1 a_AT1 distance_NN1 d_ZZ1 away_II21 from_II22 the_AT first_MD sheet_NN1 (_( fig._NN1 2.14_MC )_) ._. 
Then_RT the_AT electric_JJ fields_NN2 due_II21 to_II22 the_AT two_MC sheets_NN2 add_VV0 in_RL21 between_RL22 the_AT sheets_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC cancel_VV0 outside_II the_AT sheets_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT equipotential_JJ surfaces_NN2 are_VBR obviously_RR planes_NN2 ._. 
We_PPIS2 get_VV0 ,_, by_II integrating_VVG eqn_NN1 (_( 2.54_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
2.6_MC ._. 
The_AT parallel-plate_JJ capacitor_NN1 We_PPIS2 have_VH0 so_RG far_RR considered_VVN point_NN1 charges_NN2 ,_, dipoles_NN2 ,_, line_NN1 charges_NN2 ,_, and_CC sheet_NN1 charges_NN2 without_IW enquiring_VVG into_II the_AT problem_NN1 how_RRQ such_DA configurations_NN2 of_IO charges_NN2 come_VV0 about_RP ._. 
Indeed_RR if_CS one_PN1 gives_VVZ a_AT1 little_JJ thought_NN1 to_II the_AT matter_NN1 it_PPH1 becomes_VVZ distinctly_RR doubtful_JJ that_CST one_PN1 could_VM ever_RR establish_VVI anything_PN1 even_RR vaguely_RR resembling_VVG a_AT1 sheet_NN1 of_IO charge_NN1 ._. 
Like_II charges_NN2 repel_VV0 each_PPX221 other_PPX222 so_RR the_AT charges_NN2 constituting_VVG the_AT sheet_NN1 will_VM fly_VVI apart_RL ._. 
Interestingly_RR ,_, there_EX is_VBZ a_AT1 way_NN1 of_IO realizing_VVG sheets_NN2 of_IO charges_NN2 ._. 
We_PPIS2 just_RR need_VV0 to_TO apply_VVI a_AT1 voltage_NN1 between_II two_MC metal_NN1 plates_NN2 ._. 
What_DDQ happens_VVZ inside_II metal_NN1 plates_NN2 in_II31 response_II32 to_II33 an_AT1 applied_JJ potential_NN1 or_CC for_IF that_DD1 matter_NN1 what_DDQ happens_VVZ to_II charges_NN2 inside_II any_DD material_NN1 ?_? 
There_EX is_VBZ no_AT escape_NN1 ;_; I_PPIS1 have_VH0 to_TO say_VVI a_AT1 few_DA2 words_NN2 about_II the_AT properties_NN2 of_IO materials_NN2 ._. 
A_AT1 piece_NN1 of_IO material_NN1 contains_VVZ equal_JJ number_NN1 of_IO positive_JJ and_CC negative_JJ charges_NN2 ._. 
If_CS this_DD1 was_VBDZ not_XX so_RR there_EX would_VM be_VBI large_JJ forces_NN2 between_II various_JJ pieces_NN2 of_IO materials_NN2 ._. 
To_TO give_VVI you_PPY a_AT1 feeling_NN1 how_RRQ large_JJ these_DD2 forces_NN2 could_VM be_VBI I_PPIS1 quote_VV0 from_II Feynman_NP1 's_GE lectures_NN2 :_: &quot;_" If_CS you_PPY were_VBDR standing_VVG at_II arm_NN1 's_GE length_NN1 from_II someone_PN1 and_CC each_DD1 of_IO you_PPY had_VHD one_MC1 per_NNU21 cent_NNU22 more_DAR electrons_NN2 than_CSN protons_NN2 ,_, the_AT repelling_JJ force_NN1 would_VM be_VBI incredible_JJ ._. 
How_RGQ great_JJ ?_? 
Enough_RR to_TO lift_VVI the_AT Empire_NP1 State_NP1 Building_NN1 ?_? 
No_UH !_! 
To_TO lift_VVI Mount_NNL1 Everest_NP1 ?_? 
No_UH !_! 
The_AT repulsion_NN1 would_VM be_VBI enough_DD to_TO lift_VVI a_AT1 &quot;_" weight_NN1 &quot;_" equal_JJ to_II that_DD1 of_IO the_AT entire_JJ earth_NN1 !_! 
&quot;_" So_RR as_CSA I_PPIS1 said_VVD before_II a_AT1 piece_NN1 of_IO material_NN1 contains_VVZ equal_JJ numbers_NN2 of_IO positive_JJ and_CC negative_JJ charges_NN2 ._. 
For_IF a_AT1 class_NN1 of_IO materials_NN2 called_VVN conductors_NN2 the_AT internal_JJ charge_NN1 distribution_NN1 may_VM be_VBI looked_VVN upon_II as_II a_AT1 cloud_NN1 of_IO mobile_JJ negative_JJ particles_NN2 in_II the_AT background_NN1 of_IO immobile_JJ positive_JJ lattice_NN1 ions_NN2 ._. 
When_CS the_AT voltage_NN1 is_VBZ applied_VVN ,_, there_EX is_VBZ initially_RR an_AT1 electric_JJ field_NN1 inside_II the_AT conductor_NN1 and_CC the_AT charges_NN2 start_VV0 to_TO move_VVI under_II its_APPGE influence_NN1 ._. 
So_RR to_TO begin_VVI with_IW this_DD1 is_VBZ not_XX a_AT1 static_JJ problem_NN1 at_RR21 all_RR22 ._. 
But_CCB if_CS we_PPIS2 wait_VV0 patiently_RR for_IF a_AT1 few_DA2 picoseconds_NNT2 until_CS the_AT charges_NN2 rearrange_VV0 themselves_PPX2 there_RL will_VM be_VBI no_AT further_JJR motion_NN1 and_CC the_AT problem_NN1 belongs_VVZ to_II the_AT realm_NN1 of_IO electrostatics_NN2 ._. 
But_CCB where_RRQ will_VM the_AT electrons_NN2 find_VVI their_APPGE equilibrium_NN1 positions_NN2 ?_? 
How_RGQ far_RR will_VM they_PPHS2 move_VVI under_II the_AT influence_NN1 of_IO an_AT1 attractive_JJ electric_JJ field_NN1 ?_? 
Disregarding_VVG the_AT cases_NN2 when_RRQ the_AT electric_JJ field_NN1 is_VBZ very_RG large_JJ or_CC the_AT conductor_NN1 is_VBZ very_RG hot_JJ (_( beyond_II the_AT scope_NN1 of_IO this_DD1 course_NN1 )_) the_AT electrons_NN2 can_VM not_XX get_VVI farther_RRR than_CSN the_AT boundary_NN1 of_IO the_AT conductor_NN1 ._. 
So_RR there_EX will_VM be_VBI an_AT1 accumulation_NN1 of_IO electrons_NN2 on_II the_AT surface_NN1 of_IO one_MC1 of_IO the_AT plates_NN2 and_CC consequently_RR a_AT1 deficiency_NN1 of_IO electrons_NN2 at_II the_AT other_JJ plate_NN1 's_GE surface_NN1 ._. 
Inside_II the_AT conductor_NN1 there_EX will_VM be_VBI no_AT imbalance_NN1 of_IO charge_NN1 ._. 
This_DD1 is_VBZ an_AT1 important_JJ thing_NN1 to_TO remember_VVI ._. 
If_CS we_PPIS2 apply_VV0 a_AT1 constant_JJ voltage_NN1 then_RT ,_, after_CS the_AT elapse_NN1 of_IO a_AT1 short_JJ time_NNT1 ,_, charge_NN1 neutrality_NN1 is_VBZ re-established_VVN in_II the_AT interior_NN1 of_IO the_AT conductor_NN1 but_CCB there_EX will_VM be_VBI some_DD uncompensated_JJ charges_NN2 in_II the_AT immediate_JJ vicinity_NN1 of_IO the_AT surfaces_NN2 ._. 
How_RGQ close_RR to_II the_AT surfaces_NN2 ?_? 
It_PPH1 does_VDZ n't_XX really_RR matter_VVI ._. 
The_AT scales_NN2 are_VBR certainly_RR atomic_JJ ,_, so_CS we_PPIS2 are_VBR entitled_VVN to_TO regard_VVI these_DD2 charges_NN2 at_II the_AT surfaces_NN2 as_CSA having_VHG spread_VVN out_RP in_II two_MC dimensions_NN2 only_RR ._. 
So_RR the_AT introduction_NN1 of_IO a_AT1 surface_NN1 charge_NN1 is_VBZ not_XX unrealistic_JJ at_RR21 all_RR22 ._. 
Well_RR then_RT ,_, two_MC infinitely_RR large_JJ metal_NN1 plates_NN2 to_II which_DDQ a_AT1 constant_JJ voltage_NN1 is_VBZ applied_VVN are_VBR equivalent_JJ to_II two_MC oppositely_RR charged_VVN sheets_NN2 ._. 
The_AT electric_JJ field_NN1 between_II the_AT plates_NN2 is_VBZ then_RT &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC the_AT potential_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Consequently_RR the_AT potential_JJ difference_NN1 (_( i.e._REX the_AT voltage_NN1 )_) between_II the_AT plates_NN2 comes_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Next_MD let_VV0 us_PPIO2 work_VVI out_RP the_AT capacitance_NN1 per_II unit_NN1 surface_NN1 area_NN1 ._. 
Recall_VV0 the_AT definition_NN1 from_II circuit_NN1 theory_NN1 :_: the_AT capacitance_NN1 is_VBZ the_AT proportionality_NN1 factor_NN1 relating_VVG the_AT charge_NN1 stored_VVN on_II one_MC1 of_IO the_AT plates_NN2 to_II the_AT applied_JJ voltage_NN1 ,_, i.e._REX &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT total_JJ charge_NN1 per_II unit_NN1 surface_NN1 area_NN1 is_VBZ ps_NN2 ,_, which_DDQ leads_VVZ to_II the_AT following_JJ formula_NN1 for_IF the_AT capacitance_NN1 per_II unit_NN1 surface_NN1 area_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 shall_VM keep_VVI now_RT the_AT voltage_NN1 constant_NN1 and_CC insert_VVI a_AT1 piece_NN1 of_IO dielectric_JJ between_II the_AT plates_NN2 as_CSA shown_VVN in_II Fig._NN1 2.15_MC ._. 
What_DDQ sort_NN1 of_IO difference_NN1 will_VM that_DD1 make_VVI ?_? 
Since_CS the_AT voltage_NN1 is_VBZ the_AT same_DA and_CC the_AT dielectric_JJ is_VBZ homogeneous_JJ the_AT electric_JJ field_NN1 is_VBZ still_RR given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT flux_NN1 density_NN1 ,_, on_II the_AT other_JJ hand_NN1 ,_, will_VM be_VBI different_JJ on_II31 account_II32 of_II33 the_AT &lsqb;_( formula_NN1 &rsqb;_) relationship_NN1 ._. 
What_DDQ about_II the_AT surface-charge_JJ density_NN1 ?_? 
That_DD1 will_VM also_RR increase_VVI by_II the_AT same_DA factor_NN1 Er_FU ._. 
In_II fact_NN1 the_AT surface-charge_JJ density_NN1 will_VM be_VBI equal_JJ to_II D._NP1 We_PPIS2 can_VM easily_RR provide_VVI the_AT proof_NN1 by_II choosing_VVG a_AT1 Gaussian_JJ surface_NN1 as_CSA shown_VVN in_II Fig._NN1 2.15_MC ._. 
The_AT contribution_NN1 of_IO S-_NN1 to_II the_AT integral_JJ is_VBZ now_RT zero_MC (_( because_CS both_RR E_ZZ1 and_CC D_ZZ1 are_VBR zero_MC inside_II the_AT conductor_NN1 )_) and_CC that_DD1 of_IO S+_FO comes_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR the_AT application_NN1 of_IO Gauss_NP1 's_GE law_NN1 yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
So_RR we_PPIS2 have_VH0 proved_VVN that_CST the_AT surface-charge_JJ density_NN1 increases_VVZ by_II a_AT1 factor_NN1 Er_FU ._. 
Why_RRQ ?_? 
This_DD1 is_VBZ a_AT1 problem_NN1 that_CST rightfully_RR belongs_VVZ to_II the_AT subject_NN1 of_IO the_AT electrical_JJ properties_NN2 of_IO materials_NN2 ,_, so_CS I_PPIS1 can_VM not_XX say_VVI very_RG much_DA1 about_II it_PPH1 ._. 
In_II a_AT1 perfect_JJ dielectric_JJ there_EX are_VBR no_AT mobile_JJ charges_NN2 but_CCB lots_PN of_IO bound_JJ charges_NN2 ,_, positive_JJ and_CC negative_JJ ._. 
In_II31 response_II32 to_II33 the_AT electric_JJ field_NN1 in_II which_DDQ these_DD2 charges_NN2 find_VV0 themselves_PPX2 ,_, the_AT positive_JJ and_CC negative_JJ charges_NN2 will_VM slightly_RR separate_VVI ._. 
The_AT effect_NN1 will_VM be_VBI some_DD uncompensated_JJ charges_NN2 at_II the_AT edge_NN1 of_IO the_AT dielectric_JJ (_( Fig._NN1 2.16_MC )_) which_DDQ will_VM draw_VVI some_DD further_JJR charges_NN2 of_IO the_AT opposite_JJ sign_NN1 from_II the_AT interior_NN1 of_IO the_AT conductor_NN1 to_II its_APPGE surface_NN1 ._. 
This_DD1 is_VBZ the_AT physical_JJ mechanism_NN1 responsible_JJ for_IF the_AT increase_NN1 of_IO surface_NN1 charge_NN1 density_NN1 in_II the_AT presence_NN1 of_IO a_AT1 dielectric_JJ ._. 
Have_VH0 n't_XX we_PPIS2 made_VVN a_AT1 mistake_NN1 ?_? 
When_CS working_VVG out_RP the_AT total_JJ charge_NN1 within_II the_AT Gaussian_JJ surface_NN1 (_( the_AT right-hand_JJ side_NN1 of_IO eqn_NN1 (_( 2.63_MC we_PPIS2 ignored_VVD the_AT charge_NN1 density_NN1 on_II the_AT surface_NN1 of_IO the_AT dielectric_JJ ._. 
Surely_RR ,_, we_PPIS2 should_VM have_VHI taken_VVN that_DD1 into_II account_NN1 ._. 
What_DDQ counts_VVZ is_VBZ the_AT net_JJ surface-charge_JJ density_NN1 inside_II our_APPGE chosen_JJ volume_NN1 and_CC that_DD1 has_VHZ n't_XX increased_VVN at_RR21 all_RR22 ._. 
All_DB that_CST happened_VVD is_VBZ that_CST the_AT electric_JJ field_NN1 drew_VVD some_DD charges_NN2 to_II the_AT surface_NN1 of_IO the_AT dielectric_JJ which_DDQ caused_VVD then_RT some_DD additional_JJ charges_NN2 to_TO appear_VVI on_II the_AT surface_NN1 of_IO the_AT conductor_NN1 ._. 
So_RR the_AT surface-charge_JJ density_NN1 increased_VVN on_II the_AT surface_NN1 of_IO the_AT conductor_NN1 but_CCB not_XX the_AT net_JJ charge_NN1 density_NN1 inside_II the_AT Gaussian_JJ surface_NN1 ._. 
Have_VH0 we_PPIS2 or_CC have_VH0 we_PPIS2 not_XX made_VVN a_AT1 mistake_NN1 ?_? 
The_AT answer_NN1 depends_VVZ on_II our_APPGE definition_NN1 of_IO charge_NN1 and_CC charge_NN1 density_NN1 ._. 
If_CS q_ZZ1 in_II Gauss_NP1 's_GE law_NN1 (_( eqn_NN1 (_( 2.15_MC )_) )_) means_VVZ free_JJ charge_NN1 then_RT we_PPIS2 are_VBR all_RR21 right_RR22 ,_, then_RT we_PPIS2 can_VM ignore_VVI the_AT bound_JJ charge_NN1 on_II the_AT dielectric_JJ ._. 
In_II fact_NN1 this_DD1 is_VBZ the_AT whole_JJ point_NN1 of_IO introducing_VVG D._NP1 By_II assigning_VVG a_AT1 relative_JJ permittivity_NN1 to_II a_AT1 dielectric_JJ material_NN1 all_DB these_DD2 problems_NN2 have_VH0 been_VBN taken_VVN care_NN1 of_IO ._. 
We_PPIS2 need_VV0 to_TO consider_VVI the_AT free_JJ charges_NN2 only_RR ._. 
Let_VM21 's_VM22 not_XX forget_VVI that_CST we_PPIS2 are_VBR concerned_JJ with_IW parallel-plate_JJ capacitors_NN2 and_CC we_PPIS2 are_VBR interested_JJ in_II determining_VVG the_AT capacitance_NN1 ._. 
Since_CS the_AT insertion_NN1 of_IO the_AT dielectric_JJ increased_VVD the_AT surface_NN1 charge_NN1 density_NN1 on_II the_AT plates_NN2 by_II a_AT1 factor_NN1 with_IW the_AT voltage_NN1 remaining_VVG unchanged_JJ ,_, we_PPIS2 may_VM conclude_VVI that_CST the_AT capacitance_NN1 has_VHZ also_RR increased_VVN by_II the_AT same_DA factor_NN1 ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ._. 
Next_MD ,_, we_PPIS2 shall_VM consider_VVI a_AT1 more_RGR complicated_JJ problem_NN1 where_CS the_AT space_NN1 between_II the_AT plates_NN2 is_VBZ filled_VVN by_II two_MC different_JJ dielectrics_NN2 ,_, as_CSA shown_VVN in_II Fig._NN1 2.17_MC ._. 
Introducing_VVG subscripts_NN2 1_MC1 and_CC 2_MC for_IF denoting_VVG our_APPGE quantities_NN2 in_II the_AT two_MC dielectrics_NN2 ,_, we_PPIS2 may_VM write_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
What_DDQ else_RR do_VD0 we_PPIS2 know_VVI ?_? 
In_II each_DD1 section_NN1 the_AT relationship_NN1 (_( 2.62_MC )_) must_VM still_RR be_VBI valid_JJ ,_, so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC since_CS potential_NN1 is_VBZ additive_JJ &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 need_VV0 one_MC1 more_DAR equation_NN1 between_II our_APPGE variables_NN2 that_CST can_VM again_RT be_VBI provided_VVN by_II Gauss_NP1 's_GE law_NN1 ._. 
Using_VVG the_AT Gaussian_JJ surface_NN1 shown_VVN in_II Fig._NN1 2.17_MC and_CC noting_VVG that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT right-hand_JJ side_NN1 is_VBZ zero_MC since_CS there_EX are_VBR no_AT free_JJ charges_NN2 inside_II the_AT Gaussian_JJ surface_NN1 ._. 
Hence_RR &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC ,_, in_II31 view_II32 of_II33 eqn_NN1 (_( 2.65_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
With_IW the_AT aid_NN1 of_IO eqns_NN2 (_( 2.67_MC )_) and_CC (_( 2.71_MC )_) we_PPIS2 may_VM now_RT obtain_VVI the_AT capacitance_NN1 per_II unit_NN1 area_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) ,_, in_II agreement_NN1 with_IW the_AT tenets_NN2 of_IO circuit_NN1 theory._NNU 2.7_MC ._. 
Two-dimensional_JJ problems_NN2 We_PPIS2 have_VH0 so_RG far_RR had_VHN infinite_JJ metal_NN1 plates_NN2 and_CC infinite_JJ dielectrics_NN2 ._. 
They_PPHS2 were_VBDR chosen_VVN to_TO be_VBI infinitely_RR large_JJ in_BCL21 order_BCL22 to_TO reduce_VVI the_AT problem_NN1 to_II a_AT1 one-dimensional_JJ one_PN1 ._. 
Let_VV0 us_PPIO2 take_VVI now_RT the_AT bold_JJ step_NN1 of_IO increasing_VVG the_AT number_NN1 of_IO dimensions_NN2 by_II one_PN1 ._. 
What_DDQ is_VBZ the_AT simplest_JJT two-dimensional_JJ problem_NN1 involving_VVG conductors_NN2 ?_? 
Two_MC concentric_JJ circular_JJ cylinders_NN2 (_( Fig._NN1 2.18_MC )_) to_II which_DDQ a_AT1 voltage_NN1 is_VBZ applied_VVN and_CC where_CS the_AT medium_NN1 between_II the_AT cylinders_NN2 is_VBZ a_AT1 vacuum_NN1 ._. 
Our_APPGE aim_NN1 is_VBZ first_MD to_TO determine_VVI the_AT variation_NN1 of_IO electric_JJ field_NN1 as_II a_AT1 function_NN1 of_IO radius_NN1 and_CC then_RT to_TO work_VVI out_RP the_AT capacitance_NN1 per_II unit_NN1 length_NN1 ._. 
As_CSA you_PPY are_VBR getting_VVG used_JJ to_II it_PPH1 by_RT21 now_RT22 we_PPIS2 shall_VM start_VVI with_IW Gauss_NP1 's_GE law_NN1 ._. 
The_AT Gaussian_JJ surface_NN1 will_VM be_VBI a_AT1 cylinder_NN1 at_II radius_NN1 R_ZZ1 as_CSA shown_VVN in_II Fig._NN1 2.19_MC ._. 
Owing_II21 to_II22 a_AT1 circular_JJ symmetry_NN1 the_AT electric_JJ field_NN1 will_VM have_VHI only_RR a_AT1 radical_JJ component_NN1 ,_, independent_JJ of_IO the_AT azimuth_NN1 angle_NN1 ._. 
Hence_RR Gauss_NP1 's_GE law_NN1 (_( for_IF unit_NN1 length_NN1 of_IO cylinder_NN1 )_) takes_VVZ the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
From_II the_AT definition_NN1 C=q/V_FU we_PPIS2 get_VV0 immediately_RR for_IF the_AT capacitance_NN1 per_II unit_NN1 length_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
What_DDQ have_VH0 we_PPIS2 learned_VVN from_II the_AT solution_NN1 of_IO our_APPGE first_MD two-dimensional_JJ problem_NN1 involving_VVG conductors_NN2 ?_? 
We_PPIS2 have_VH0 found_VVN that_CST the_AT electric_JJ field_NN1 varies_VVZ as_II 1/R_FU ._. 
There_EX is_VBZ nothing_PN1 new_JJ in_II that_DD1 ;_; we_PPIS2 came_VVD to_II the_AT same_DA conclusion_NN1 earlier_RRR when_CS studying_VVG the_AT field_NN1 of_IO a_AT1 line_NN1 charge_NN1 ._. 
We_PPIS2 have_VH0 managed_VVN ,_, though_CS ,_, to_TO derive_VVI the_AT capacitance_NN1 of_IO concentric_JJ cylinders_NN2 ,_, a_AT1 formula_NN1 used_VVN in_II practical_JJ engineering_NN1 ,_, so_CS we_PPIS2 have_VH0 certainly_RR achieved_VVN something_PN1 ._. 
It_PPH1 would_VM now_RT be_VBI easy_JJ to_TO go_VVI on_RP and_CC work_VVI out_RP the_AT field_NN1 between_II two_MC concentric_JJ spheres_NN2 ._. 
We_PPIS2 shall_VM however_RR resist_VVI the_AT temptation_NN1 ._. 
We_PPIS2 would_VM learn_VVI little_RR ,_, because_CS by_II exploiting_VVG spherical_JJ symmetry_NN1 we_PPIS2 would_VM just_RR have_VHI to_TO deal_VVI with_IW another_DD1 pseudo-one-dimensional_JJ problem_NN1 ._. 
It_PPH1 is_VBZ important_JJ of_RR21 course_RR22 that_CST the_AT field_NN1 varies_VVZ as_II 1/r2_FU in_II that_DD1 case_NN1 ,_, and_CC it_PPH1 is_VBZ also_RR of_IO some_DD use_NN1 to_TO know_VVI the_AT formula_NN1 for_IF the_AT capacitance_NN1 of_IO a_AT1 spherical_JJ capacitor_NN1 but_CCB you_PPY can_VM work_VVI that_DD1 out_RP yourself_PPX1 if_CS you_PPY are_VBR interested_JJ ._. 
Let_VV0 us_PPIO2 look_VVI at_II a_AT1 real_JJ two-dimensional_JJ problem_NN1 instead_RR ._. 
We_PPIS2 shall_VM take_VVI two_MC conducting_VVG cylinders_NN2 of_IO radius_NN1 a_AT1 (_( Fig._NN1 2.20_MC )_) and_CC apply_VV0 a_AT1 voltage_NN1 between_II them_PPHO2 ._. 
How_RRQ can_VM we_PPIS2 find_VVI the_AT electric_JJ field_NN1 ?_? 
That_DD1 should_VM be_VBI easy_JJ ._. 
For_IF calculating_VVG the_AT field_NN1 in_II point_NN1 P_ZZ1 we_PPIS2 need_VV0 two_MC Gaussian_JJ surfaces_NN2 ,_, namely_REX cylinders_NN2 R1_FO and_CC R2_FO ,_, as_CSA shown_VVN in_II Fig._NN1 2.20_MC ._. 
Then_RT owing_II21 to_II22 the_AT charge_NN1 on_II one_MC1 cylinder_NN1 (_( per_II unit_NN1 length_NN1 of_RR21 course_RR22 )_) &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC owing_II21 to_II22 the_AT charge_NN1 on_II the_AT other_JJ cylinder_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT problem_NN1 is_VBZ linear_JJ ,_, superposition_NN1 is_VBZ permissible_JJ ,_, so_RR all_DB we_PPIS2 need_VV0 to_TO do_VDI is_VBZ to_TO add_VVI vectorially_RR E1_FO ,_, and_CC E2_FO and_CC the_AT field_NN1 at_II point_NN1 P_ZZ1 is_VBZ determined_VVN ._. 
Unfortunately_RR the_AT method_NN1 is_VBZ wrong_JJ ._. 
Why_RRQ ?_? 
Gauss_NP1 's_GE law_NN1 is_VBZ valid_JJ ,_, and_CC afterwards_RT we_PPIS2 did_VDD no_AT more_DAR than_CSN added_VVD the_AT field_NN1 due_II21 to_II22 the_AT two_MC charges_NN2 ._. 
This_DD1 is_VBZ permissible_JJ indeed_RR if_CS the_AT charges_NN2 are_VBR at_II fixed_JJ positions_NN2 ._. 
In_II the_AT absence_NN1 of_IO cylinder_NN1 1_MC1 the_AT field_NN1 due_II21 to_II22 the_AT charge_NN1 on_II cylinder_NN1 2_MC is_VBZ correctly_RR given_VVN by_II eqn_NN1 (_( 2.79_MC )_) ._. 
But_CCB we_PPIS2 have_VH0 used_VVN cylindrical_JJ symmetry_NN1 ._. 
We_PPIS2 have_VH0 relied_VVN on_II the_AT fact_NN1 that_CST the_AT charge_NN1 distribution_NN1 on_II cylinder_NN1 1_MC1 is_VBZ uniform_JJ ._. 
When_CS we_PPIS2 put_VV0 cylinder_NN1 2_MC there_RL ,_, the_AT circular_JJ symmetry_NN1 is_VBZ broken_VVN ._. 
The_AT negative_JJ charge_NN1 on_II cylinder_NN1 2_MC will_VM attract_VVI the_AT positive_JJ charge_NN1 on_II cylinder_NN1 1_MC1 therefore_RR the_AT part_NN1 of_IO cylinder_NN1 1_MC1 facing_JJ cylinder_NN1 2_MC will_VM have_VHI a_AT1 higher_JJR surface-charge_JJ density_NN1 than_CSN the_AT opposite_JJ side_NN1 ._. 
Gauss_NP1 's_GE law_NN1 is_VBZ still_RR valid_JJ by_II eqns_NN2 (_( 2.79_MC )_) and_CC (_( 2.80_MC )_) do_VD0 not_XX follow_VVI from_II it_PPH1 ._. 
How_RRQ can_VM we_PPIS2 find_VVI a_AT1 solution_NN1 ?_? 
Well_RR ,_, in_II this_DD1 particular_JJ case_NN1 we_PPIS2 can_VM find_VVI the_AT solution_NN1 by_II attacking_VVG another_DD1 problem_NN1 :_: that_DD1 of_IO two_MC line_NN1 charges_NN2 as_CSA shown_VVN in_II Fig._NN1 2.21_MC ._. 
The_AT potential_NN1 for_IF one_MC1 line_NN1 charge_NN1 was_VBDZ given_VVN by_II eqn_NN1 (_( 2.45_MC )_) ._. 
For_IF two_MC line_NN1 charges_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT equipotential_JJ surfaces_NN2 are_VBR given_VVN by_II the_AT equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS k_ZZ1 is_VBZ a_AT1 constant_JJ ._. 
Doing_VDG a_AT1 bit_NN1 of_IO analytical_JJ geometry_NN1 it_PPH1 turns_VVZ out_RP that_CST the_AT equipotential_JJ surfaces_NN2 are_VBR circular_JJ cylinders_NN2 as_CSA shown_VVN by_II dotted_JJ lines_NN2 in_II Fig._NN1 2.22_MC ._. 
We_PPIS2 can_VM now_RT re-state_VVI the_AT two-cylinder_JJ problem_NN1 of_IO Fig._NN1 2.20_MC as_CSA presented_VVN in_II Fig._NN1 2.23_MC ;_; we_PPIS2 need_VV0 only_RR relate_VVI the_AT parameters_NN2 a_AT1 ,_, b_ZZ1 ,_, d_ZZ1 ,_, and_CC K_ZZ1 to_II each_PPX221 other_PPX222 ._. 
The_AT calculation_NN1 outlined_VVN above_RL yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
&lsqb;_( formula_NN1 &rsqb;_) ._. 
Thus_RR the_AT solution_NN1 for_IF the_AT two-cylinder_JJ problem_NN1 is_VBZ provided_VVN by_II the_AT solution_NN1 for_IF the_AT two-line-source_JJ problem_NN1 having_VHG the_AT same_DA amount_NN1 of_IO charge_NN1 per_II unit_NN1 length_NN1 ._. 
We_PPIS2 may_VM now_RT work_VVI out_RP the_AT capacitance_NN1 if_CS we_PPIS2 wish_VV0 ._. 
The_AT potential_JJ difference_NN1 may_VM be_VBI obtained_VVN as_CSA follows_VVZ &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT capacitance_NN1 per_II unit_NN1 length_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS b&gt;a_FO we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Having_VHG got_VVN so_RG far_RR it_PPH1 might_VM be_VBI of_IO interest_NN1 to_TO work_VVI out_RP the_AT variation_NN1 of_IO surface-charge_JJ density_NN1 on_II one_MC1 of_IO the_AT cylinders_NN2 as_II a_AT1 function_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
In_BCL21 order_BCL22 to_TO obtain_VVI the_AT surface-charge_JJ density_NN1 we_PPIS2 need_VV0 to_TO work_VVI out_RP D_ZZ1 and_CC E_ZZ1 on_II the_AT surface_NN1 of_IO the_AT cylinder_NN1 ._. 
It_PPH1 is_VBZ very_RG simple_JJ in_II principle_NN1 ;_; we_PPIS2 just_RR need_VV0 to_TO use_VVI the_AT gradient_NN1 relationship_NN1 between_II &lsqb;_( formula_NN1 &rsqb;_) and_CC E_ZZ1 yielding_JJ &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT quantity_NN1 of_IO interest_NN1 is_VBZ the_AT relative_JJ variation_NN1 of_IO the_AT surface-charge_JJ density_NN1 ,_, i.e._REX &lsqb;_( formula_NN1 &rsqb;_) which_DDQ may_VM be_VBI obtained_VVN (_( after_II a_AT1 fair_JJ amount_NN1 of_IO tedious_JJ algebra_NN1 )_) in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT above_JJ equation_NN1 is_VBZ plotted_VVN in_II Fig._NN1 2.24_MC for_CS b/a=1.1_FU ,_, 1.5_MC ,_, 3_MC ,_, and_CC 10_MC ._. 
It_PPH1 may_VM be_VBI seen_VVN that_CST for_IF a_AT1 low_JJ value_NN1 of_IO b/a_FU the_AT surface-charge_JJ density_NN1 changes_NN2 considerably_RR around_II the_AT circumference_NN1 of_IO the_AT cylinder_NN1 ._. 
When_CS b/a1_FU there_EX is_VBZ hardly_RR any_DD variation_NN1 ._. 
So_RR it_PPH1 is_VBZ clear_JJ that_CST our_APPGE first_MD approach_NN1 (_( demonstrated_VVN in_II Fig._NN1 2.20_MC )_) had_VHD no_AT general_JJ validity_NN1 ._. 
But_CCB ,_, if_CS b/a1_FU the_AT charges_NN2 on_II the_AT two_MC cylinders_NN2 have_VH0 little_DA1 effect_NN1 upon_II each_PPX221 other_PPX222 ,_, and_CC the_AT approach_NN1 is_VBZ permissible_JJ ._. 
We_PPIS2 shall_VM finish_VVI off_RP the_AT two-cylinder_JJ problem_NN1 by_II working_VVG out_RP the_AT capacitance_NN1 when_CS b/a&gt;1_FU on_II the_AT basis_NN1 of_IO eqns_NN2 (_( 2.79_MC )_) and_CC (_( 2.80_MC )_) ._. 
On_II the_AT line_NN1 connecting_VVG the_AT centres_NN2 of_IO the_AT cylinders_NN2 E1_FO and_CC E2_FO add_VV0 algebraically_RR so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ,_, leading_VVG to_II a_AT1 capacitance_NN1 in_II agreement_NN1 with_IW eqn_NN1 (_( 2.88_MC )_) ._. 
Having_VHG solved_VVN the_AT two-cylinder_JJ problem_NN1 we_PPIS2 are_VBR not_XX much_RR wiser_JJR ;_; we_PPIS2 still_RR do_VD0 n't_XX know_VVI how_RRQ to_TO tackle_VVI a_AT1 general_JJ problem_NN1 ._. 
There_EX is_VBZ though_RR one_MC1 general_JJ rule_NN1 we_PPIS2 can_VM set_VVI up_RP easily_RR enough_RR ._. 
The_AT electric_JJ field_NN1 must_VM always_RR be_VBI perpendicular_JJ to_II the_AT surface_NN1 of_IO a_AT1 conductor_NN1 ._. 
The_AT reason_NN1 is_VBZ that_CST otherwise_RR the_AT surface_NN1 charges_NN2 would_VM be_VBI in_II motion_NN1 contrary_II21 to_II22 the_AT assumption_NN1 that_CST a_AT1 static_JJ equilibrium_NN1 exists_VVZ ._. 
If_CS the_AT electric_JJ field_NN1 is_VBZ perpendicular_JJ to_II a_AT1 surface_NN1 then_RT ,_, owing_II21 to_II22 the_AT &lsqb;_( formula_NN1 &rsqb;_) relationship_NN1 ,_, that_DD1 surface_NN1 must_VM be_VBI an_AT1 equipotential_JJ ._. 
So_RR if_CS we_PPIS2 work_VV0 in_II31 terms_II32 of_II33 the_AT potential_NN1 the_AT boundary_NN1 condition_NN1 may_VM be_VBI easily_RR formulated_VVN :_: &lsqb;_( formula_NN1 &rsqb;_) =_FO constant_JJ on_II all_DB conductor_NN1 surfaces_NN2 ._. 
A_AT1 further_JJR advantage_NN1 of_IO using_VVG (_( as_CSA I_PPIS1 mentioned_VVD several_DA2 times_NNT2 before_II )_) is_VBZ that_CST we_PPIS2 get_VV0 rid_VVN of_IO the_AT vectorial_JJ nature_NN1 of_IO the_AT problem_NN1 ._. 
The_AT mathematical_JJ problem_NN1 is_VBZ then_RT to_TO solve_VVI the_AT differential_JJ equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) in_II31 conjunction_II32 with_II33 the_AT boundary_NN1 condition_NN1 that_CST &lsqb;_( formula_NN1 &rsqb;_) constant_JJ on_II all_DB conductor_NN1 surfaces_NN2 ._. 
Are_VBR there_EX any_DD general_JJ methods_NN2 for_IF solving_VVG eqn_NN1 (_( 2.93_MC )_) ?_? 
There_EX are_VBR n't_XX any_DD ._. 
However_RR ,_, if_CS there_EX is_VBZ no_AT free_JJ charge_NN1 in_II the_AT space_NN1 between_II the_AT conductors_NN2 ,_, and_CC strictly_RR for_IF two-dimensional_JJ problems_NN2 ,_, there_EX is_VBZ a_AT1 method_NN1 to_II which_DDQ the_AT adjective_NN1 &quot;_" general_NN1 &quot;_" might_VM be_VBI attached_VVN ,_, a_AT1 method_NN1 that_CST provides_VVZ plenty_PN of_IO answers_NN2 but_CCB not_XX necessarily_RR to_II the_AT questions_NN2 asked_VVD ._. 
The_AT method_NN1 is_VBZ based_VVN on_II the_AT theory_NN1 of_IO complex_JJ variables_NN2 ._. 
You_PPY have_VH0 heard_VVN about_II complex_JJ variables_NN2 ,_, and_CC I_PPIS1 believe_VV0 you_PPY have_VH0 come_VVN across_II the_AT Cauchy-Riemann_NP1 relationships_NN2 ._. 
So_RR you_PPY know_VV0 that_CST if_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 complex_JJ variable_NN1 and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 complex_JJ function_NN1 ,_, then_RT the_AT relationships_NN2 &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) hold_VV0 ._. 
Differentiating_JJ eqn_NN1 (_( 2.96_MC )_) with_II31 respect_II32 to_II33 x_ZZ1 and_CC eqn_NN1 (_( 2.97_MC )_) with_II31 respect_II32 to_II33 y_ZZ1 we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, whereas_CS differentiation_NN1 in_II the_AT opposite_JJ order_NN1 leads_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR both_RR u_ZZ1 and_CC v_ZZ1 are_VBR solutions_NN2 of_IO Laplace_NP1 's_GE equation_NN1 ._. 
This_DD1 is_VBZ sheer_JJ luck_NN1 to_TO find_VVI a_AT1 solution_NN1 so_RG easily_RR ._. 
In_II fact_NN1 we_PPIS2 are_VBR even_RR luckier_JJR ._. 
It_PPH1 turns_VVZ out_RP (_( though_CS I_PPIS1 am_VBM not_XX going_VVGK to_TO prove_VVI it_PPH1 here_RL )_) that_CST the_AT v_ZZ1 (_( x_ZZ1 ,_, y_ZZ1 )_) =_FO constant_JJ curves_NN2 are_VBR orthogonal_JJ trajectories_NN2 of_IO the_AT u_ZZ1 (_( x_ZZ1 ,_, y_ZZ1 )_) =_FO constant_JJ curves_NN2 ._. 
Thus_RR if_CS we_PPIS2 identify_VV0 one_MC1 of_IO the_AT functions_NN2 with_IW the_AT potential_NN1 ,_, the_AT other_JJ one_PN1 will_VM represent_VVI the_AT field_NN1 lines_NN2 ._. 
It_PPH1 is_VBZ really_RR as_RG simple_JJ as_CSA that_DD1 ._. 
Take_VV0 any_DD reasonable_JJ function_NN1 of_IO a_AT1 complex_JJ variable_NN1 and_CC we_PPIS2 have_VH0 the_AT solution_NN1 of_IO an_AT1 electrostatic_JJ problem_NN1 ._. 
What_DDQ is_VBZ the_AT simplest_JJT function_NN1 we_PPIS2 can_VM take_VVI ?_? 
Probably_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
Take_VV0 u1_FO =_FO x1_FO and_CC u2_FO =_FO x2_FO as_CSA conductor_NN1 surfaces_NN2 (_( Fig._NN1 2.25_MC )_) then_RT the_AT y_ZZ1 =_FO constant_JJ lines_NN2 will_VM represent_VVI the_AT field_NN1 lines_NN2 ._. 
This_DD1 example_NN1 does_VDZ not_XX offer_VVI anything_PN1 new_JJ (_( we_PPIS2 just_RR got_VVD the_AT configuration_NN1 of_IO an_AT1 infinitely_RR large_JJ parallel-plate_JJ capacitor_NN1 )_) ,_, but_CCB we_PPIS2 can_VM see_VVI that_CST the_AT method_NN1 works_VVZ ._. 
Let_VV0 us_PPIO2 try_VVI as_CSA our_APPGE next_MD example_NN1 something_PN1 more_RGR difficult_JJ like_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Identifying_VVG now_RT v=2xy_FO with_IW the_AT potential_NN1 and_CC u=x2-y2_FO with_IW the_AT field_NN1 lines_NN2 we_PPIS2 get_VV0 two_MC sets_NN2 of_IO orthogonal_JJ hyperbolas_NN2 ._. 
Taking_VVG for_REX21 example_REX22 v_ZZ1 =_FO 1_MC1 and_CC 2_MC for_IF the_AT conductor_NN1 surfaces_NN2 we_PPIS2 have_VH0 solved_VVN an_AT1 electrostatic_JJ problem_NN1 as_CSA may_VM be_VBI seen_VVN in_II Fig._NN1 2.26_MC ._. 
Would_VM anyone_PN1 ever_RR have_VHI to_TO do_VDI anything_PN1 with_IW infinite_JJ conductors_NN2 shaped_VVN like_II that_DD1 ?_? 
Very_RG unlikely_JJ ._. 
But_CCB a_AT1 finite_JJ corner_NN1 is_VBZ of_IO interest_NN1 ,_, after_II all_DB we_PPIS2 often_RR have_VH0 electric_JJ fields_NN2 in_II metal_NN1 boxes_NN2 ._. 
So_RR we_PPIS2 can_VM take_VVI v_ZZ1 =_FO 0_MC (_( giving_VVG the_AT asymptotes_NN2 )_) as_CSA one_MC1 of_IO the_AT conductors_NN2 and_CC the_AT ensuing_JJ picture_NN1 (_( Fig._NN1 2.27_MC )_) does_VDZ indeed_RR give_VVI some_DD intuitive_JJ &quot;_" feel_VV0 &quot;_" for_IF the_AT electric_JJ field_NN1 lines_NN2 in_II the_AT vicinity_NN1 of_IO a_AT1 corner_NN1 ._. 
The_AT electric_JJ field_NN1 itself_PPX1 may_VM be_VBI found_VVN from_II the_AT equations_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS you_PPY are_VBR fond_JJ of_IO mathematical_JJ games_NN2 there_EX are_VBR plenty_PN of_IO functions_NN2 to_TO experiment_VVI with_IW ._. 
You_PPY can_VM be_VBI sure_JJ that_CST you_PPY will_VM find_VVI a_AT1 solution_NN1 of_IO Laplace_NP1 's_GE equation_NN1 satisfying_VVG the_AT boundary_NN1 conditions_NN2 ,_, but_CCB you_PPY will_VM have_VHI to_TO find_VVI out_RP which_DDQ problem_NN1 you_PPY have_VH0 obtained_VVN the_AT solution_NN1 of_IO ._. 
I_PPIS1 will_VM not_XX dwell_VVI much_RR longer_RRR on_II conformal_JJ mapping_NN1 (_( this_DD1 is_VBZ incidentally_RR the_AT term_NN1 most_RGT often_RR used_VVN for_IF describing_VVG the_AT method_NN1 )_) but_CCB will_VM give_VVI a_AT1 few_DA2 more_DAR examples_NN2 ._. 
Let_VV0 us_PPIO2 place_VVI two_MC coplanar_JJ conducting_NN1 plates_NN2 very_RG close_JJ to_II each_PPX221 other_PPX222 (_( Fig._NN1 2.28_MC )_) and_CC apply_VV0 a_AT1 potential_JJ difference_NN1 between_II them_PPHO2 ._. 
At_II some_DD distance_NN1 away_II21 from_II22 the_AT gap_NN1 the_AT equipotentials_NN2 will_VM be_VBI planes_NN2 and_CC the_AT field_NN1 lines_NN2 will_VM be_VBI circles_NN2 ._. 
In_II the_AT immediate_JJ vicinity_NN1 of_IO the_AT gap_NN1 the_AT situation_NN1 is_VBZ more_RGR complicated_JJ ,_, as_CSA shown_VVN in_II Fig._NN1 2.29_MC ._. 
The_AT equipotentials_NN2 are_VBR confocal_JJ hyperbolas_NN2 and_CC the_AT electric_JJ lines_NN2 of_IO force_NN1 are_VBR given_VVN by_II confocal_JJ ellipses_NN2 ._. 
Fig._NN1 2.30_MC shows_VVZ the_AT field_NN1 lines_NN2 and_CC equipotentials_NN2 for_IF two_MC semi-infinite_JJ parallel_NN1 conducting_VVG plates_NN2 raised_VVN to_II different_JJ potentials_NN2 ._. 
It_PPH1 tells_VVZ us_PPIO2 what_DDQ happens_VVZ at_II the_AT edge_NN1 of_IO a_AT1 capacitor_NN1 and_CC can_VM also_RR give_VVI a_AT1 numerical_JJ estimate_NN1 of_IO the_AT scattered_JJ capacitance_NN1 (_( by_II which_DDQ the_AT capacitance_NN1 of_IO a_AT1 real_JJ capacitor_NN1 differs_VVZ from_II that_DD1 worked_VVD out_RP on_II the_AT basis_NN1 of_IO the_AT infinite-plate_JJ model_NN1 )_) ._. 
The_AT foregoing_JJ three_MC examples_NN2 provided_VVD further_JJR illustrations_NN2 of_IO the_AT usefulness_NN1 of_IO complex_JJ functions_NN2 for_IF solving_VVG electrostatic_JJ problems_NN2 (_( for_IF further_JJR information_NN1 on_II these_DD2 mappings_NN2 see_VV0 ,_, for_REX21 example_REX22 ,_, Simonyi_NP1 (_( 1963_MC )_) )_) ._. 
For_IF axially_RR symmetric_JJ cases_NN2 the_AT method_NN1 is_VBZ unfortunately_RR not_XX applicable_JJ ._. 
I_PPIS1 shall_VM not_XX be_VBI discussing_VVG here_RL the_AT methods_NN2 that_CST are_VBR applicable_JJ but_CCB shall_VM present_VVI the_AT solution_NN1 to_II one_MC1 particular_JJ problem_NN1 arising_VVG in_II electron_NN1 optics_NN1 ,_, that_DD1 of_IO two_MC closely_RR spaced_VVN conducting_VVG cylinders_NN2 at_II different_JJ potentials_NN2 (_( Fig._NN1 2.31_MC )_) ._. 
2.8_MC ._. 
The_AT method_NN1 of_IO images_NN2 This_DD1 is_VBZ again_RT a_AT1 rather_RG specific_JJ method_NN1 suitable_JJ for_IF the_AT solution_NN1 of_IO a_AT1 limited_JJ class_NN1 of_IO problems_NN2 in_II both_DB2 two_MC and_CC three_MC dimensions_NN2 ._. 
The_AT simplest_JJT way_NN1 of_IO introducing_VVG the_AT method_NN1 is_VBZ to_TO present_VVI again_RT the_AT field_NN1 lines_NN2 and_CC the_AT equipotential_JJ surfaces_NN2 for_IF two_MC equal_JJ charges_NN2 of_IO opposite_JJ sign_NN1 (_( Fig._NN1 2.4_MC )_) ._. 
Let_VV0 us_PPIO2 now_RT place_VVI an_AT1 infinite_JJ conductor_NN1 plane_NN1 halfway_RR between_II the_AT charges_NN2 as_CSA shown_VVN in_II Fig._NN1 2.32_MC ._. 
Since_CS the_AT y_ZZ1 =_FO 0_MC plane_NN1 was_VBDZ an_AT1 equipotential_JJ surface_NN1 anyway_RR (_( the_AT field_NN1 lines_NN2 were_VBDR perpendicular_JJ to_II it_PPH1 )_) ,_, nothing_PN1 changes_VVZ ._. 
Hence_RR we_PPIS2 have_VH0 found_VVN the_AT solution_NN1 for_IF a_AT1 charge_NN1 in_II31 front_II32 of_II33 an_AT1 infinite_JJ plane_NN1 ._. 
Working_VVG backwards_RL we_PPIS2 may_VM now_RT say_VVI that_CST the_AT effect_NN1 of_IO an_AT1 infinite_JJ conducting_NN1 plane_NN1 is_VBZ equivalent_JJ to_II that_DD1 of_IO a_AT1 charge_NN1 of_IO opposite_JJ sign_NN1 placed_VVN in_II the_AT mirror_NN1 position_NN1 (_( the_AT negative_JJ charge_NN1 is_VBZ the_AT image_NN1 of_IO the_AT positive_JJ charge_NN1 in_II the_AT plane_NN1 )_) ._. 
As_II an_AT1 example_NN1 let_VV0 us_PPIO2 work_VVI out_RP the_AT surface-charge_JJ density_NN1 on_II the_AT surface_NN1 of_IO the_AT conducting_NN1 plane_NN1 due_II21 to_II22 a_AT1 positive_JJ charge_NN1 q_ZZ1 above_II it_PPH1 (_( Fig._NN1 2.33_MC (_( a_ZZ1 )_) )_) ._. 
This_DD1 is_VBZ a_AT1 difficult_JJ boundary-value_JJ problem_NN1 ,_, but_CCB using_VVG the_AT method_NN1 of_IO images_NN2 we_PPIS2 only_RR need_VV0 to_TO determine_VVI the_AT electric_JJ flux_NN1 density_NN1 due_II21 to_II22 the_AT two_MC point_NN1 charges_NN2 ._. 
According_II21 to_II22 Fig._NN1 2.33(b)_FO the_AT electric_JJ field_NN1 at_II a_AT1 distance_NN1 r_ZZ1 from_II the_AT charge_NN1 is_VBZ given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) (_( the_AT negative_JJ sign_NN1 is_VBZ due_II21 to_II22 the_AT fact_NN1 that_CST the_AT direction_NN1 of_IO the_AT electric_JJ field_NN1 is_VBZ opposite_JJ to_II the_AT normal_JJ to_II the_AT plane_NN1 )_) ,_, and_CC consequently_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
There_EX are_VBR various_JJ generalizations_NN2 of_IO the_AT method_NN1 ._. 
A_AT1 charge_NN1 in_II a_AT1 corner_NN1 has_VHZ three_MC images_NN2 (_( Fig._NN1 2.34_MC )_) but_CCB there_EX are_VBR as_RG many_DA2 as_CSA five_MC images_NN2 in_II a_AT1 wedge_NN1 of_IO 60_MC (_( Fig._NN1 2.35_MC )_) ._. 
You_PPY can_VM now_RT work_VVI out_RP for_IF yourself_PPX1 how_RRQ many_DA2 images_NN2 a_AT1 charge_NN1 in_II a_AT1 wedge_NN1 of_IO angle_NN1 &lsqb;_( formula_NN1 &rsqb;_) has_VHZ (_( unfortunately_RR n_ZZ1 must_VM be_VBI an_AT1 integer_NN1 )_) ._. 
The_AT essence_NN1 of_IO the_AT method_NN1 is_VBZ that_CST a_AT1 conductor_NN1 may_VM be_VBI replaced_VVN by_II a_AT1 set_NN1 of_IO charges_NN2 without_IW any_DD change_NN1 in_II the_AT pattern_NN1 of_IO field_NN1 lines_NN2 and_CC equipotentials_NN2 ._. 
Another_DD1 example_NN1 of_IO a_AT1 mirror_NN1 charge_NN1 ,_, this_DD1 time_NNT1 in_II a_AT1 cylinder_NN1 ,_, is_VBZ provided_VVN by_II Figs_NN2 2.22_MC and_CC 2.23_MC ,_, redrawn_VVN in_II a_AT1 more_RGR suitable_JJ form_NN1 in_II Fig._NN1 2.36_MC ._. 
This_DD1 is_VBZ an_AT1 example_NN1 we_PPIS2 have_VH0 already_RR worked_VVN out_RP ._. 
According_II21 to_II22 our_APPGE new_JJ interpretation_NN1 the_AT line_NN1 charge_NN1 p1_FO has_VHZ its_APPGE mirror_NN1 charge_NN1 -p1_NN1 at_II a_AT1 distance_NN1 2d_NNU ._. 
It_PPH1 is_VBZ also_RR possible_JJ to_TO define_VVI a_AT1 mirror_NN1 charge_NN1 in_II a_AT1 conducting_NN1 sphere_NN1 and_CC in_II some_DD purely_RR dielectric_JJ configurations_NN2 but_CCB we_PPIS2 will_VM not_XX discuss_VVI them_PPHO2 here._NNU 2.9_MC ._. 
Dielectric_JJ boundaries_NN2 The_AT conditions_NN2 for_IF conducting_VVG boundaries_NN2 have_VH0 been_VBN simple_JJ enough_RR ._. 
Unfortunately_RR not_XX all_DB boundaries_NN2 are_VBR comprised_VVN of_IO conductors_NN2 ;_; some_DD of_IO them_PPHO2 are_VBR dielectrics_NN2 ,_, and_CC even_RR worse_RRR we_PPIS2 have_VH0 to_TO consider_VVI sometimes_RT imperfect_JJ dielectrics_NN2 that_CST have_VH0 a_AT1 finite_JJ conductivity_NN1 ._. 
The_AT proper_JJ boundary_NN1 conditions_NN2 may_VM be_VBI derived_VVN in_II the_AT usual_JJ manner_NN1 with_IW the_AT aid_NN1 of_IO Gauss_NP1 's_GE law_NN1 ._. 
In_II the_AT general_JJ case_NN1 the_AT boundaries_NN2 are_VBR neither_RR planes_NN2 nor_CC circles_NN2 ,_, but_CCB this_DD1 fact_NN1 does_VDZ not_XX need_VVI to_TO bother_VVI us_PPIO2 ._. 
If_CS we_PPIS2 investigate_VV0 a_AT1 small_JJ enough_DD part_NN1 of_IO the_AT boundary_NN1 between_II two_MC arbitrary_JJ media_NN we_PPIS2 can_VM always_RR regard_VVI the_AT boundary_NN1 as_II a_AT1 plane_NN1 surface_NN1 ._. 
The_AT Gaussian_JJ surface_NN1 may_VM then_RT be_VBI chosen_VVN in_II the_AT form_NN1 of_IO a_AT1 cylinder_NN1 ,_, as_CSA shown_VVN in_II Fig._NN1 2.37_MC ._. 
If_CS the_AT height_NN1 of_IO the_AT cylinder_NN1 dh_NNU is_VBZ approaching_VVG zero_NN1 then_RT the_AT total_JJ flux_NN1 is_VBZ going_VVG through_II the_AT top_NN1 and_CC bottom_JJ surfaces_NN2 ,_, i.e._REX &lsqb;_( formula_NN1 &rsqb;_) ._. 
Keeping_VVG the_AT treatment_NN1 entirely_RR general_JJ we_PPIS2 shall_VM permit_VVI now_RT the_AT presence_NN1 of_IO a_AT1 surface_NN1 charge_NN1 (_( made_VVN up_RP of_IO free_JJ charges_NN2 ;_; it_PPH1 is_VBZ still_RR true_JJ that_CST the_AT bound_JJ charges_NN2 of_IO dielectrics_NN2 do_VD0 not_XX count_VVI )_) ,_, hence_RR ,_, when_CS dh_NNU 0_MC ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC it_PPH1 follows_VVZ from_II the_AT above_JJ two_MC equations_NN2 that_CST &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ now_RT entirely_RR general_JJ ._. 
It_PPH1 includes_VVZ as_RG special_JJ cases_NN2 both_RR eqn_NN1 (_( 2.63_MC )_) (_( when_CS Dn2_FO is_VBZ zero_MC inside_II the_AT metal_NN1 )_) ,_, and_CC eqn_NN1 (_( 2.72_MC )_) (_( when_CS both_DB2 dielectrics_NN2 are_VBR perfect_JJ so_CS21 that_CS22 no_AT free_JJ charges_NN2 can_VM reside_VVI on_II the_AT boundary_NN1 surface_NN1 )_) ._. 
So_RG far_RR ,_, so_RG good_JJ ._. 
We_PPIS2 have_VH0 derived_VVN the_AT condition_NN1 for_IF the_AT normal_JJ component_NN1 of_IO D._NP1 What_DDQ about_II the_AT tangential_JJ component_NN1 ?_? 
We_PPIS2 can_VM get_VVI that_DD1 by_II taking_VVG this_DD1 time_NNT1 the_AT line_NN1 integral_JJ of_IO the_AT electric_JJ field_NN1 along_II both_DB2 sides_NN2 of_IO the_AT boundary_NN1 as_CSA shown_VVN in_II Fig._NN1 2.38_MC ._. 
Assuming_VVG that_CST dl_MC 0_MC and_CC noting_VVG that_CST for_IF a_AT1 closed_JJ contour_NN1 the_AT line_NN1 integral_JJ of_IO the_AT electric_JJ field_NN1 vanishes_VVZ we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, i.e._REX ,_, the_AT tangential_JJ component_NN1 of_IO the_AT electric_JJ field_NN1 strength_NN1 is_VBZ constant_JJ across_II a_AT1 boundary_NN1 ._. 
This_DD1 means_VVZ that_CST the_AT electric_JJ field_NN1 will_VM refract_VVI at_II the_AT boundary_NN1 of_IO two_MC dielectrics_NN2 ,_, as_CSA shown_VVN in_II Fig._NN1 2.39_MC ._. 
The_AT relevant_JJ equations_NN2 are_VBR &lsqb;_( formula_NN1 &rsqb;_) ._. 
Thus_RR unless_CS the_AT incident_NN1 angle_NN1 is_VBZ 90_MC the_AT field_NN1 lines_NN2 will_VM have_VHI a_AT1 break_NN1 at_II the_AT boundary_NN1 of_IO two_MC dielectrics_NN2 ._. 
An_AT1 example_NN1 is_VBZ shown_VVN in_II Fig._NN1 2.40_MC ._. 
We_PPIS2 have_VH0 a_AT1 line_NN1 charge_NN1 in_II31 front_II32 of_II33 a_AT1 dielectric_JJ that_CST fills_VVZ half_DB the_AT space_NN1 ._. 
The_AT field_NN1 lines_NN2 ,_, as_CSA expected_VVN ,_, refract_VV0 when_CS entering_VVG the_AT dielectric_JJ (_( for_IF a_AT1 mathematical_JJ solution_NN1 in_II31 terms_II32 of_II33 Images_NN2 ,_, see_VV0 Clemmow_NP1 (_( 1973_MC )_) )_) ._. 
Another_DD1 example_NN1 demonstrating_VVG the_AT same_DA phenomenon_NN1 may_VM be_VBI seen_VVN in_II Fig._NN1 2.41_MC ._. 
A_AT1 dielectric_JJ sphere_NN1 inserted_VVN into_II a_AT1 homogeneous_JJ electric_JJ field_NN1 in_II air_NN1 or_CC a_AT1 vacuum_NN1 &quot;_" attracts_VVZ &quot;_" the_AT field_NN1 lines_NN2 ._. 
The_AT physical_JJ explanation_NN1 is_VBZ that_CST the_AT dielectric_JJ is_VBZ capable_JJ of_IO carrying_VVG a_AT1 higher_JJR flux_NN1 density_NN1 ._. 
The_AT mathematical_JJ solution_NN1 is_VBZ given_VVN below_RL in_II spherical_JJ coordinates_NN2 ,_, r_ZZ1 ,_, 0_MC (_( the_AT solution_NN1 is_VBZ of_RR21 course_RR22 independent_JJ of_IO the_AT azimuth_NN1 angle_NN1 &lsqb;_( formula_NN1 &rsqb;_) )_) :_: &lsqb;_( formula_NN1 &rsqb;_) ,_, from_II which_DDQ E_ZZ1 and_CC D_ZZ1 may_VM be_VBI derived_VVN (_( E0_FO is_VBZ the_AT electric_JJ field_NN1 strength_NN1 in_II the_AT absence_NN1 of_IO the_AT sphere_NN1 )_) ._. 
The_AT boundary_NN1 conditions_NN2 ,_, eqn_NN1 (_( 2.107_MC )_) ,_, are_VBR satisfied_VVN at_II r_ZZ1 =_FO a_AT1 ._. 
If_CS you_PPY are_VBR interested_JJ you_PPY can_VM check_VVI them_PPHO2 (_( Example_NN1 2.12_MC )_) ._. 
In_II an_AT1 isotropic_JJ medium_NN1 (_( in_II which_DDQ the_AT dielectric_JJ constant_NN1 is_VBZ a_AT1 scalar_JJ not_XX a_AT1 tensor_NN1 )_) the_AT directions_NN2 of_IO the_AT E_ZZ1 and_CC D_ZZ1 lines_NN2 coincide_VV0 ,_, thus_RR Figs._NN2 2.40_MC and_CC 2.41_MC may_VM refer_VVI to_II either_DD1 ._. 
However_RRQV if_CS we_PPIS2 want_VV0 to_TO interpret_VVI the_AT density_NN1 of_IO field_NN1 lines_NN2 as_CSA being_VBG proportional_JJ to_II the_AT field_NN1 strength_NN1 then_RT both_DB2 figures_NN2 must_VM represent_VVI D_ZZ1 since_CS it_PPH1 is_VBZ the_AT flux_NN1 of_IO D_ZZ1 that_CST remains_VVZ constant_JJ across_II a_AT1 dielectric_JJ boundary._NNU 2.10_MC ._. 
Electrostatic_JJ energy_NN1 Electrostatics_NN1 is_VBZ characterized_VVN by_II charges_NN2 located_VVN at_II certain_JJ positions_NN2 ._. 
If_CS we_PPIS2 know_VV0 the_AT positions_NN2 of_IO all_DB the_AT charges_NN2 ,_, we_PPIS2 know_VV0 everything_PN1 so_RR we_PPIS2 should_VM know_VVI the_AT energy_NN1 as_RR21 well_RR22 ._. 
There_EX is_VBZ unfortunately_RR no_AT obvious_JJ way_NN1 of_IO writing_VVG an_AT1 expression_NN1 for_IF the_AT energy_NN1 in_II the_AT simultaneous_JJ presence_NN1 of_IO all_DB the_AT charges_NN2 ._. 
It_PPH1 is_VBZ ,_, however_RR ,_, fairly_RR easy_JJ to_TO work_VVI out_RP the_AT energy_NN1 as_CSA we_PPIS2 bring_VV0 in_RP the_AT charges_NN2 from_II infinity_NN1 one_MC1 by_II one_MC1 ._. 
The_AT first_MD charge_NN1 is_VBZ brought_VVN in_RP without_IW any_DD opposition_NN1 ._. 
The_AT energy_NN1 of_IO one_MC1 charge_NN1 alone_RR is_VBZ zero_MC ._. 
The_AT second_MD charge_NN1 is_VBZ brought_VVN in_RP in_II the_AT presence_NN1 of_IO the_AT first_MD one_PN1 ._. 
The_AT work_NN1 done_VDN is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
There_EX are_VBR now_RT two_MC charges_NN2 ,_, q1_FO and_CC q2_FO ,_, and_CC we_PPIS2 bring_VV0 a_AT1 third_MD one_PN1 ,_, q3_FO ,_, from_II infinity_NN1 ._. 
The_AT work_NN1 done_VDN is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Three_MC charges_NN2 are_VBR probably_RR enough_RR for_IF seeing_VVG the_AT general_JJ trend_NN1 so_CS we_PPIS2 shall_VM now_RT sum_VVI up_RP the_AT two_MC partial_JJ energies_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ may_VM also_RR be_VBI written_VVN in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 may_VM be_VBI recognized_VVN that_CST the_AT terms_NN2 in_II the_AT brackets_NN2 represent_VV0 the_AT potentials_NN2 due_II21 to_II22 the_AT other_JJ two_MC charges_NN2 ._. 
It_PPH1 is_VBZ easy_JJ to_TO guess_VVI that_CST in_II the_AT general_JJ case_NN1 one_PN1 would_VM have_VHI ,_, instead_RR ,_, the_AT potential_NN1 due_II21 to_II22 all_DB other_JJ charges_NN2 ._. 
Hence_RR the_AT general_JJ term_NN1 is_VBZ of_IO the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT potential_NN1 at_II the_AT point_NN1 where_RRQ q1_FO resides_VVZ (_( ignoring_VVG the_AT contribution_NN1 of_IO q1_FO )_) ._. 
The_AT expression_NN1 for_IF the_AT total_JJ energy_NN1 of_IO n_ZZ1 charges_NN2 is_VBZ then_RT &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT transition_NN1 from_II discrete_JJ to_II distributed_JJ charge_NN1 can_VM be_VBI made_VVN as_CSA follows_VVZ :_: q1_FO is_VBZ replaced_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC the_AT summation_NN1 by_II integration_NN1 ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS the_AT volume_NN1 r_ZZ1 must_VM contain_VVI all_DB the_AT charges_NN2 ._. 
Thus_RR if_CS the_AT charge_NN1 distribution_NN1 is_VBZ known_VVN we_PPIS2 should_VM first_MD determine_VVI the_AT potential_NN1 (_( from_II eqn_NN1 (_( 2.24_MC )_) and_CC then_RT we_PPIS2 may_VM use_VVI eqn_NN1 (_( 2.116_MC )_) for_IF calculating_VVG the_AT energy_NN1 ._. 
This_DD1 is_VBZ a_AT1 perfectly_RR reasonable_JJ procedure_NN1 ,_, but_CCB we_PPIS2 have_VH0 to_TO admit_VVI that_DD1 eqn_NN1 (_( 2.116_MC )_) is_VBZ rarely_RR used_VVN ._. 
It_PPH1 is_VBZ usually_RR transformed_VVN into_II another_DD1 more_RGR popular_JJ form_NN1 ._. 
Since_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC by_II eqn_NN1 (_( A.2_FO )_) &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 may_VM first_MD transform_VVI eqn_NN1 (_( 2.116_MC )_) into_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Noting_VVG further_RRR that_DD1 &lsqb;_( formula_NN1 &rsqb;_) and_CC applying_VVG Gauss_NP1 's_GE theorem_NN1 to_II the_AT first_MD term_NN1 in_II the_AT bracket_NN1 we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS S_ZZ1 is_VBZ the_AT closed_JJ surface_NN1 of_IO the_AT volume_NN1 r_ZZ1 ._. 
Now_RT this_DD1 is_VBZ another_DD1 perfectly_RR reasonable_JJ formula_NN1 to_TO use_VVI once_RR the_AT volume_NN1 is_VBZ chosen_VVN ._. 
Well_RR ,_, let_VV0 us_PPIO2 choose_VVI it_PPH1 in_II the_AT form_NN1 of_IO a_AT1 big_JJ sphere_NN1 ._. 
If_CS r_ZZ1 the_AT radius_NN1 of_IO this_DD1 sphere_NN1 is_VBZ big_JJ enough_RR then_RT the_AT potential_JJ decays_NN2 as_II 1/r_FU (_( cf._VV0 Section_NN1 2.4_MC )_) and_CC the_AT electric_JJ field_NN1 as_II 1/r2_FU ._. 
Thus_RR the_AT integrand_NN1 decays_VVZ as_CSA 1_MC1 /r3_NN1 ,_, whereas_CS the_AT surface_NN1 increases_VVZ only_RR as_CSA r2_FO ._. 
Consequently_RR the_AT surface_NN1 integral_JJ must_VM vanish_VVI as_CSA r_ZZ1 </w>_NULL ._. 
We_PPIS2 are_VBR now_RT left_VVN with_IW the_AT sought-for_JJ expression_NN1 ._. 
Provided_CS the_AT volume_NN1 integral_JJ is_VBZ over_II all_DB space_VV0 the_AT energy_NN1 is_VBZ given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ a_AT1 simple_JJ and_CC physically_RR easily_RR interpretable_JJ expression_NN1 ._. 
Wherever_RRQV there_EX is_VBZ electric_JJ field_NN1 ,_, there_EX is_VBZ energy_NN1 as_RR21 well_RR22 ._. 
So_RR how_RRQ should_VM you_PPY think_VVI about_II electrostatic_JJ energy_NN1 ?_? 
You_PPY are_VBR certainly_RR entitled_VVN to_TO think_VVI in_II31 terms_II32 of_II33 charges_NN2 and_CC potentials_NN2 ._. 
In_BCL21 order_BCL22 to_TO create_VVI a_AT1 charge_NN1 distribution_NN1 a_AT1 certain_JJ amount_NN1 of_IO work_NN1 has_VHZ to_TO be_VBI done_VDN ,_, and_CC that_DD1 is_VBZ available_JJ to_II us_PPIO2 in_II the_AT form_NN1 of_IO electrostatic_JJ energy_NN1 ._. 
But_CCB it_PPH1 is_VBZ better_JJR (_( it_PPH1 is_VBZ more_RGR general_JJ )_) to_TO regard_VVI the_AT electric_JJ field_NN1 as_II the_AT agent_NN1 with_IW which_DDQ the_AT energy_NN1 has_VHZ been_VBN deposited_VVN ._. 
One_PN1 usually_RR says_VVZ that_CST the_AT energy_NN1 is_VBZ stored_VVN in_II the_AT electric_JJ field_NN1 ._. 
Let_VV0 us_PPIO2 work_VVI out_RP ,_, as_CSA an_AT1 example_NN1 ,_, the_AT stored_JJ energy_NN1 of_IO a_AT1 parallel-plate_JJ capacitor_NN1 ._. 
The_AT electric_JJ field_NN1 strength_NN1 is_VBZ given_VVN (_( eqn_NN1 (_( 2.57_MC )_) )_) by_II E_ZZ1 =_FO V/d_ZZ1 ,_, hence_RR the_AT stored_JJ energy_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS S0_FO is_VBZ the_AT area_NN1 of_IO the_AT capacitor_NN1 plates_NN2 ._. 
Of_RR21 course_RR22 ,_, you_PPY can_VM derive_VVI the_AT above_JJ expression_NN1 from_II circuit_NN1 theory_NN1 ,_, so_RG once_RR21 more_RR22 circuit_NN1 theory_NN1 and_CC electromagnetic_JJ theory_NN1 give_VV0 the_AT same_DA answer_NN1 ._. 
Could_VM we_PPIS2 use_VVI this_DD1 stored_JJ energy_NN1 for_IF calculating_VVG the_AT magnitude_NN1 of_IO forces_NN2 acting_VVG ,_, could_VM we_PPIS2 get_VVI the_AT direction_NN1 of_IO the_AT forces_NN2 ?_? 
Take_VV0 the_AT following_JJ example_NN1 from_II mechanics_NN2 :_: a_AT1 mass_JJ m_ZZ1 at_II rest_NN1 at_II a_AT1 height_NN1 h_ZZ1 has_VHZ a_AT1 potential_JJ energy_NN1 mgh_NNU ._. 
In_II which_DDQ direction_NN1 will_VM it_PPH1 move_VVI if_CSW its_APPGE support_NN1 is_VBZ taken_VVN away_RL ?_? 
It_PPH1 will_VM try_VVI to_TO reduce_VVI the_AT height_NN1 h_ZZ1 in_BCL21 order_BCL22 to_TO minimize_VVI its_APPGE potential_JJ energy_NN1 ._. 
While_CS the_AT energy_NN1 of_IO the_AT capacitor_NN1 increases_VVZ that_DD1 of_IO the_AT battery_NN1 decreases_VVZ ._. 
Why_RRQ ?_? 
Because_CS an_AT1 increase_NN1 of_IO dC_NN1 in_II the_AT capacitance_NN1 requires_VVZ an_AT1 amount_NN1 of_IO extra_JJ charge_NN1 dq_NNU =_FO VdC_NP1 ,_, which_DDQ must_VM flow_VVI from_II the_AT battery_NN1 to_II the_AT capacitor_NN1 ._. 
So_RR the_AT loss_NN1 of_IO energy_NN1 by_II the_AT battery_NN1 is_VBZ V_ZZ1 dq_NNU =_FO V2_FO dC_NN1 ._. 
Hence_RR the_AT net_JJ gain_NN1 of_IO energy_NN1 of_IO the_AT system_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, in_II agreement_NN1 with_IW eqn_NN1 (_( 2.124_MC )_) ._. 
So_RR if_CS we_PPIS2 consider_VV0 the_AT capacitor_NN1 and_CC the_AT battery_NN1 together_RL the_AT total_JJ energy_NN1 of_IO the_AT system_NN1 decreases_VVZ as_II the_AT capacitance_NN1 increases_NN2 ._. 
We_PPIS2 obtain_VV0 the_AT same_DA force_NN1 between_II the_AT plates_NN2 whether_CSW the_AT battery_NN1 is_VBZ disconnected_VVN or_CC not_XX as_CSA we_PPIS2 should_VM ._. 
Examples_NN2 2_MC 1_MC1 ._. 
Four_MC point_NN1 charges_NN2 of_IO equal_JJ magnitude_NN1 are_VBR located_VVN at_II the_AT corners_NN2 of_IO a_AT1 square_JJ as_CSA shown_VVN in_II Fig._NN1 2.42_MC ._. 
Determine_VV0 the_AT magnitude_NN1 and_CC direction_NN1 of_IO the_AT force_NN1 on_II each_DD1 charge._NNU 2.2_MC ._. 
A_AT1 cloud_NN1 of_IO charged_JJ particles_NN2 having_VHG a_AT1 total_JJ charge_NN1 q_ZZ1 ,_, fills_VVZ uniformly_RR the_AT volume_NN1 of_IO a_AT1 sphere_NN1 of_IO radius_NN1 a_AT1 ._. 
Find_VV0 the_AT electric_JJ field_NN1 at_II a_AT1 distance_NN1 r_ZZ1 from_II the_AT centre_NN1 of_IO the_AT cloud_NN1 both_RR for_IF r&lt;a_FO and_CC r&gt;a._FO 2.3_MC ._. 
Determine_VV0 the_AT electric_JJ field_NN1 on_II the_AT axis_NN1 of_IO a_AT1 charged_JJ ring_NN1 of_IO radius_NN1 a_AT1 carrying_VVG a_AT1 uniform_JJ line_NN1 charge_NN1 of_IO p1_FO coulomb_VV0 per_II unit_NN1 length._NNU 2.4_MC ._. 
A_AT1 linear_JJ line_NN1 charge_NN1 of_IO pl_NN1 coulomb_NN1 per_II unit_NN1 length_NN1 extends_VVZ from_II the_AT origin_NN1 to_II z_ZZ1 =_FO </w>_NULL ._. 
Determine_VV0 the_AT electric_JJ field_NN1 at_II an_AT1 arbitrary_JJ point_NN1 using_VVG cylindrical_JJ coordinates_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
2.5_MC ._. 
Determine_VV0 the_AT electric_JJ field_NN1 as_II a_AT1 function_NN1 of_IO r_ZZ1 and_CC 0_MC produced_VVN by_II the_AT axial_JJ quadrupole_NN1 shown_VVN in_II Fig._NN1 2.43_MC ._. 
Assume_VV0 that_DD1 rd._NN1 2.6_MC ._. 
An_AT1 infinitely_RR long_RR ,_, perfectly_RR conducting_VVG cylinder_NN1 of_IO radius_NN1 a_AT1 is_VBZ placed_VVN into_II a_AT1 uniform_JJ electric_JJ field_NN1 perpendicularly_RR to_II the_AT direction_NN1 of_IO the_AT field_NN1 lines_NN2 ._. 
The_AT potential_JJ function_NN1 for_IF this_DD1 case_NN1 is_VBZ given_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS A_ZZ1 is_VBZ a_AT1 constant_JJ and_CC R_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) are_VBR polar_JJ coordinates_NN2 centred_VVN at_II the_AT axis_NN1 of_IO the_AT cylinder._NNU (_( i_ZZ1 )_) Show_VV0 that_CST the_AT resultant_JJ electric_JJ field_NN1 satisfies_VVZ the_AT boundary_NN1 conditions_NN2 ,_, (_( ii_MC )_) Determine_VV0 the_AT differential_JJ equation_NN1 of_IO the_AT field_NN1 lines._NNU 2.7_MC ._. 
Show_VV0 that_CST the_AT potential_JJ function_NN1 given_VVN in_II Example_NN1 6_MC satisfies_VVZ Laplace_NP1 's_GE equation_NN1 in_II cylindrical_JJ coordinates._NNU 2.8_MC ._. 
Two_MC point_NN1 charges_NN2 q_ZZ1 and_CC -pq_NN1 (_( &lsqb;_( formula_NN1 &rsqb;_) )_) are_VBR placed_VVN at_II the_AT points_NN2 (_( 0,0,0_MC )_) and_CC (_( 0,0_MC ,_, d_ZZ1 )_) of_IO a_AT1 cartesian_JJ coordinate_NN1 system_NN1 ._. 
Find_VV0 the_AT zero_NN1 potential_NN1 surface._NNU 2.9_MC ._. 
Determine_VV0 the_AT force_NN1 upon_II a_AT1 point_NN1 charge_NN1 placed_VVN inside_II a_AT1 conducting_NN1 sphere_NN1 at_II a_AT1 distance_NN1 a_AT1 from_II the_AT centre_NN1 of_IO the_AT sphere_NN1 ._. 
(_( Hint_NN1 :_: Find_VV0 the_AT mirror_NN1 charge_NN1 in_II the_AT sphere_NN1 based_VVN on_II the_AT calculations_NN2 of_IO the_AT previous_JJ example._NNU )_) 2.10_MC ._. 
Determine_VV0 the_AT capacitance_NN1 per_II unit_NN1 length_NN1 of_IO a_AT1 two-wire_JJ transmission_NN1 line_NN1 above_II a_AT1 perfectly_RR conducting_VVG earth_NN1 (_( Fig._NN1 2.44_MC )_) ._. 
Assume_VV0 that_CST d1_FO ,_, d2D._FO 2.11_MC Derive_VV0 eqns_NN2 (_( 2.83_MC )_) -(2.85)._NNU 2.12_MC Show_VV0 that_DD1 eqn_NN1 (_( 2.110_MC )_) satisfies_VVZ the_AT boundary_NN1 conditions_NN2 at_II r_ZZ1 =_FO a._NNU 2.13_MC Two_MC long_JJ concentric_JJ cylinders_NN2 of_IO radii_NN2 a_AT1 and_CC b_ZZ1 are_VBR separated_VVN by_II a_AT1 dielectric_JJ of_IO relative_JJ permittivity_NN1 Er_FU ._. 
They_PPHS2 are_VBR held_VVN with_IW their_APPGE common_JJ axis_NN1 vertical_NN1 ,_, so_BCL21 as_BCL22 to_TO provide_VVI a_AT1 liquid-level_JJ gauge_NN1 ._. 
If_CS the_AT capacitance_NN1 of_IO the_AT gauge_NN1 is_VBZ C_ZZ1 when_RRQ a_AT1 fraction_NN1 a_AT1 of_IO its_APPGE length_NN1 is_VBZ immersed_VVN ,_, find_VV0 how_RRQ its_APPGE sensitivity_NN1 &lsqb;_( formula_NN1 &rsqb;_) varies_VVZ with_IW a_AT1 and_CC with_IW the_AT relative_JJ permittivity_NN1 of_IO the_AT fluid._NNU 2.14_MC ._. 
Find_VV0 the_AT equipotentials_NN2 and_CC lines_NN2 of_IO force_NN1 represented_VVN by_II the_AT following_JJ conformal_JJ transformations_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
2.15_MC ._. 
A_AT1 capacitor_NN1 made_VVN of_IO concentric_JJ cylinders_NN2 has_VHZ an_AT1 inner_JJ radius_NN1 a_AT1 ,_, outer_JJ radius_NN1 b_ZZ1 ,_, and_CC length_NN1 l._NNU it_PPH1 is_VBZ filled_VVN with_IW a_AT1 dielectric_JJ of_IO relative_JJ permittivity_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Derive_VV0 an_AT1 expression_NN1 for_IF the_AT maximum_JJ stored_JJ energy_NN1 ,_, W_ZZ1 max_NN1 considering_CS21 that_CS22 the_AT dielectric_JJ breaks_NN2 down_RP at_II a_AT1 field_NN1 intensity_NN1 Eb_NP1 ._. 
Calculate_VV0 W_ZZ1 max_NN1 for_IF the_AT case_NN1 when_CS a_ZZ1 =_FO 5mm_NNU ,_, b_ZZ1 =_FO 10mm_NNU ,_, l_ZZ1 =_FO 100mm_NNU ,_, Eb_NP1 =_FO 2_MC 107_MC V_ZZ1 m-1_FO ,_, Er_FU =_FO 2.25._MC 2.16_MC ._. 
Find_VV0 the_AT lateral_JJ force_NN1 on_II the_AT dielectric_JJ slab_NN1 partially_RR filling_VVG the_AT space_NN1 between_II two_MC parallel_JJ capacitor_NN1 plates_NN2 (_( Fig._NN1 2.45_MC )_) having_VHG a_AT1 voltage_NN1 V_ZZ1 between_II them._NNU 2.17_MC ._. 
One_MC1 plate_NN1 of_IO a_AT1 parallel-plate_JJ capacitor_NN1 having_VHG an_AT1 area_NN1 S_ZZ1 is_VBZ suspended_VVN at_II its_APPGE centre_NN1 from_II a_AT1 spring_NN1 of_IO stiffness_NN1 k_ZZ1 ._. 
The_AT other_JJ plate_NN1 is_VBZ held_VVN fixed_JJ at_II a_AT1 distance_NN1 d_ZZ1 ._. 
What_DDQ is_VBZ the_AT minimum_JJ voltage_NN1 to_TO be_VBI applied_VVN between_II the_AT plates_NN2 for_IF pulling_VVG them_PPHO2 together_RL ?_? 2.18_MC ._. 
A_AT1 variable_JJ capacitor_NN1 consists_VVZ of_IO two_MC sets_NN2 of_IO n_ZZ1 interconnected_JJ semicircular_JJ conducting_NN1 plates_NN2 which_DDQ can_VM be_VBI given_VVN mutual_JJ rotation_NN1 as_CSA shown_VVN in_II Fig._NN1 2.46_MC ._. 
Derive_VV0 expressions_NN2 for_IF the_AT torque_NN1 needed_VVD to_TO hold_VVI 0_MC constant_JJ under_II the_AT following_JJ conditions_NN2 :_: (_( i_ZZ1 )_) when_RRQ the_AT plates_NN2 are_VBR connected_VVN to_II a_AT1 source_NN1 of_IO constant_JJ voltage_NN1 V_ZZ1 ;_; (_( ii_MC )_) after_II the_AT plates_NN2 have_VH0 been_VBN rotated_VVN to_II their_APPGE position_NN1 of_IO maximum_JJ capacitance_NN1 ,_, charged_VVN to_II voltage_NN1 V_ZZ1 ,_, disconnected_VVD from_II the_AT supply_NN1 and_CC then_RT rotated_VVN to_II a_AT1 new_JJ position_NN1 ._. 
What_DDQ limits_VVZ the_AT maximum_JJ voltage_NN1 that_CST can_VM be_VBI achieved_VVN by_II the_AT operations_NN2 described_VVN in_II (_( ii_MC )_) ?_? 3_MC ._. 
Steady_JJ currents_NN2 3.1_MC ._. 
The_AT basic_JJ equations_NN2 IN_II this_DD1 chapter_NN1 we_PPIS2 shall_VM be_VBI concerned_JJ with_IW phenomena_NN2 in_II which_DDQ the_AT main_JJ role_NN1 is_VBZ played_VVN by_II the_AT current_NN1 of_IO charged_JJ particles_NN2 ._. 
All_DB our_APPGE variables_NN2 can_VM be_VBI functions_NN2 of_IO space_NN1 but_CCB are_VBR independent_JJ of_IO time_NNT1 ._. 
It_PPH1 is_VBZ easy_JJ to_TO present_VVI the_AT relevant_JJ equations_NN2 ;_; we_PPIS2 have_VH0 all_DB of_IO eqns_NN2 (_( 1.1_MC )_) -(1.7)_JJ but_CCB have_VH0 to_TO substitute_VVI a/at_FU =_FO O_ZZ1 ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ._. 
There_EX is_VBZ a_AT1 difficulty_NN1 with_IW eqn_NN1 (_( 3.6_MC )_) ._. 
It_PPH1 is_VBZ correct_JJ for_IF most_DAT of_IO the_AT chapter_NN1 with_IW &lsqb;_( formula_NN1 &rsqb;_) taken_VVN as_II a_AT1 constant_JJ but_CCB breaks_VVZ down_RP for_IF ferromagnetic_JJ materials_NN2 ,_, which_DDQ will_VM be_VBI discussed_VVN in_II Section_NN1 3.11_MC ._. 
All_DB the_AT other_JJ equations_NN2 are_VBR all_RR21 right_RR22 but_CCB will_VM not_XX necessarily_RR provide_VVI the_AT simplest_JJT starting_NN1 point_NN1 for_IF solving_VVG a_AT1 given_JJ problem_NN1 ._. 
We_PPIS2 shall_VM ,_, therefore_RR ,_, introduce_VVI a_AT1 number_NN1 of_IO alternative_JJ formulations_NN2 ._. 
First_MD ,_, we_PPIS2 could_VM search_VVI for_IF some_DD analogue_NN1 of_IO the_AT potential_JJ function_NN1 which_DDQ proved_VVD so_RG useful_JJ in_II electrostatics_NN1 ._. 
We_PPIS2 found_VVD there_RL that_CST by_II choosing_VVG a_AT1 scalar_JJ function_NN1 in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 could_VM automatically_RR satisfy_VVI the_AT other_JJ equation_NN1 for_IF the_AT electric_JJ field_NN1 strength_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
Can_VM we_PPIS2 do_VDI the_AT same_DA thing_NN1 for_IF the_AT magnetic_JJ quantities_NN2 ?_? 
No_UH ,_, but_CCB we_PPIS2 can_VM do_VDI a_AT1 similar_JJ thing_NN1 ._. 
Since_CS it_PPH1 is_VBZ the_AT equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) that_CST needs_VVZ to_TO be_VBI satisfied_VVN ,_, we_PPIS2 should_VM choose_VVI the_AT potential_NN1 as_II a_AT1 vector_NN1 (_( called_VVN ,_, not_XX without_IW logic_NN1 ,_, the_AT vector_NN1 potential_NN1 )_) ,_, defined_VVN by_II the_AT equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Substituting_VVG the_AT above_JJ equation_NN1 into_II eqn_NN1 (_( 3.1_MC )_) we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, or_CC using_VVG the_AT vector_NN1 relation_NN1 (_( eqn_NN1 (_( A.6_FO )_) )_) we_PPIS2 obtain_VV0 the_AT modified_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 equation_NN1 can_VM be_VBI further_RRR simplified_VVN to_II &lsqb;_( formula_NN1 &rsqb;_) by_II choosing_NN1 (_( in_II the_AT physicist_NN1 's_GE jargon_NN1 this_DD1 is_VBZ called_VVN choosing_VVG the_AT gauge_NN1 )_) &lsqb;_( formula_NN1 &rsqb;_) ._. 
Can_VM we_PPIS2 do_VDI that_DD1 ?_? 
One_PN1 ca_VM n't_XX usually_RR assign_VVI some_DD arbitrary_JJ value_NN1 to_II the_AT divergence_NN1 of_IO a_AT1 vector_NN1 function_NN1 ._. 
In_II the_AT present_JJ case_NN1 ,_, however_RR ,_, we_PPIS2 do_VD0 have_VHI some_DD freedom_NN1 of_IO choice_NN1 ._. 
The_AT definition_NN1 of_IO A_ZZ1 by_II eqn_NN1 (_( 3.8_MC )_) is_VBZ not_XX unique_JJ ._. 
If_CS we_PPIS2 add_VV0 to_II A_ZZ1 the_AT gradient_NN1 of_IO a_AT1 scalar_JJ function_NN1 ,_, the_AT resulting_JJ vector_NN1 A'_NP1 still_RR gives_VVZ the_AT same_DA magnetic_JJ field_NN1 because_CS &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR a_AT1 suitable_JJ choice_NN1 of_IO will_NN1 ensure_VVI that_CST &lsqb;_( formula_NN1 &rsqb;_) ._. 
So_RR we_PPIS2 are_VBR left_VVN with_IW eqn_NN1 (_( 3.11_MC )_) ._. 
If_CS the_AT current_JJ density_NN1 is_VBZ specified_VVN ,_, eqn_NN1 (_( 3.11_MC )_) will_VM provide_VVI the_AT solution_NN1 for_IF A_ZZ1 from_II which_DDQ B_ZZ1 can_VM be_VBI determined_VVN ._. 
How_RRQ can_VM we_PPIS2 find_VVI a_AT1 solution_NN1 for_IF A_ZZ1 ?_? 
There_EX is_VBZ nothing_PN1 easier_RRR ._. 
We_PPIS2 only_RR need_VV0 to_TO remember_VVI the_AT differential_JJ equation_NN1 for_IF the_AT scalar_JJ potential_NN1 &lsqb;_( formula_NN1 &rsqb;_) (_( eqn_NN1 (_( 2.12_MC )_) )_) and_CC its_APPGE solution_NN1 in_II the_AT form_NN1 of_IO eqn_NN1 (_( 2.24_MC )_) ._. 
By_II analogy_NN1 the_AT general_JJ solution_NN1 of_IO eqn_NN1 (_( 3.11_MC )_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
A_AT1 simple_JJ integration_NN1 will_VM yield_VVI A_ZZ1 if_CS J_ZZ1 is_VBZ given_VVN ._. 
We_PPIS2 shall_VM go_VVI through_II a_AT1 number_NN1 of_IO examples_NN2 later_RRR ._. 
For_IF the_AT moment_NN1 let_VV0 us_PPIO2 use_VVI the_AT above_JJ expression_NN1 for_IF deriving_VVG Biot-Savart_NP1 's_GE law_NN1 ._. 
First_MD we_PPIS2 shall_VM assume_VVI that_CST the_AT current_JJ density_NN1 is_VBZ confined_VVN to_II a_AT1 thin_JJ wire_NN1 in_II which_DDQ case_VV0 the_AT integration_NN1 variable_NN1 may_VM be_VBI changed_VVN to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS S_ZZ1 is_VBZ a_AT1 vector_NN1 normal_JJ to_II the_AT cross-section_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT area_NN1 of_IO the_AT cross-section_NN1 ,_, and_CC dI_FW is_VBZ an_AT1 elementary_JJ vector_NN1 along_II the_AT tangent_NN1 of_IO the_AT wire_NN1 ._. 
Noting_VVG further_RRR that_DD1 J.S_NP2 =_FO I_ZZ1 and_CC that_CST I_PPIS1 must_VM be_VBI constant_JJ along_II the_AT wire_NN1 we_PPIS2 may_VM write_VVI eqn_NN1 (_( 3.15_MC )_) in_II the_AT modified_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, whence_RRQ the_AT magnetic_JJ field_NN1 strength_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 have_VH0 to_TO stop_VVI here_RL for_IF a_AT1 moment_NN1 to_TO sort_VVI out_RP the_AT coordinates_NN2 ._. 
As_CSA may_VM be_VBI seen_VVN in_II Fig._NN1 3.1_MC the_AT coordinates_NN2 of_IO the_AT wire_NN1 element_NN1 are_VBR x'_JJ ,_, y'_PPY ,_, z'_VV0 ,_, whereas_CS the_AT coordinates_NN2 of_IO the_AT point_NN1 where_RRQ we_PPIS2 wish_VV0 to_TO determine_VVI the_AT magnetic_JJ field_NN1 are_VBR set_VVN of_IO coordinates_NN2 ._. 
Thus_RR the_AT curl_NN1 operates_VVZ on_II the_AT coordinates_NN2 of_IO P_ZZ1 but_CCB not_XX on_II those_DD2 of_IO dl_MC leading_JJ to_II &lsqb;_( formula_NN1 &rsqb;_) where_RRQ ir_NN1 is_VBZ the_AT unit_NN1 vector_NN1 in_II the_AT direction_NN1 r_ZZ1 and_CC we_PPIS2 have_VH0 made_VVN use_NN1 of_IO the_AT vector_NN1 relation_NN1 (_( A3_FO )_) in_II the_AT Appendix_NN1 ._. 
This_DD1 is_VBZ Biot-Savart_NP1 's_GE law_NN1 derived_VVD directly_RR from_II Maxwell_NP1 's_GE equations_NN2 ._. 
Are_VBR there_EX any_DD other_JJ ways_NN2 of_IO determining_VVG the_AT magnetic_JJ field_NN1 ?_? 
Well_RR ,_, one_PN1 can_VM use_VVI eqn_NN1 (_( 3.1_MC )_) as_CSA it_PPH1 is_VBZ ,_, but_CCB very_RG often_RR one_PN1 is_VBZ better_JJR off_RP by_II using_VVG its_APPGE integral_JJ form_NN1 that_CST can_VM be_VBI obtained_VVN by_II integrating_VVG both_DB2 sides_NN2 of_IO eqn_NN1 (_( 3.1_MC )_) over_II a_AT1 surface_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
By_II using_VVG Stokes_NP1 '_GE theorem_NN1 for_IF the_AT left-hand-side_NN1 and_CC recognizing_VVG that_CST the_AT integral_JJ of_IO the_AT current_JJ density_NN1 gives_VVZ the_AT current_JJ ,_, the_AT above_JJ equation_NN1 reduces_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS the_AT line_NN1 integration_NN1 is_VBZ along_II the_AT curve_NN1 C_ZZ1 enclosing_VVG the_AT surface_NN1 ._. 
The_AT positive_JJ sense_NN1 of_IO the_AT integration_NN1 path_NN1 is_VBZ defined_VVN relative_II21 to_II22 the_AT positive_JJ current_JJ direction_NN1 according_II21 to_II22 the_AT usual_JJ right-hand_JJ convention_NN1 ._. 
Equation_NN1 3.21_MC )_) is_VBZ known_VVN as_II Ampre_NP1 's_GE law_NN1 ._. 
We_PPIS2 have_VH0 now_RT a_AT1 number_NN1 of_IO equations_NN2 describing_VVG the_AT same_DA thing_NN1 ._. 
Which_DDQ equation_NN1 should_VM we_PPIS2 use_VVI in_II a_AT1 practical_JJ case_NN1 ,_, the_AT equation_NN1 for_IF the_AT vector_NN1 potential_NN1 ,_, Ampre_NP1 's_GE law_NN1 ,_, Biot-Savart_NP1 's_GE law_NN1 ,_, or_CC attack_VV0 directly_RR Maxwell_NP1 's_GE equations_NN2 ?_? 
It_PPH1 's_VBZ hard_JJ to_TO tell_VVI ._. 
Experience_VV0 with_IW practical_JJ calculations_NN2 helps_VVZ ,_, but_CCB even_RR after_II years_NNT2 of_IO work_NN1 one_PN1 usually_RR remains_VVZ this_DD1 side_NN1 of_IO infallibility_NN1 ._. 
I_PPIS1 regret_VV0 to_TO say_VVI that_CST there_EX are_VBR no_AT general_JJ guidelines_NN2 available_JJ ._. 
If_CS one_PN1 manages_VVZ to_TO find_VVI the_AT simplest_JJT method_NN1 at_II the_AT first_MD attempt_NN1 that_CST can_VM mostly_RR be_VBI attributed_VVN to_II good_JJ luck_NN1 ._. 
Can_VM we_PPIS2 claim_VVI then_RT that_CST the_AT vector_NN1 potential_NN1 is_VBZ as_RG useful_JJ as_CSA its_APPGE scalar_JJ counterpart_NN1 in_II electrostatics_NN1 ?_? 
For_IF computational_JJ purposes_NN2 the_AT answer_NN1 is_VBZ an_AT1 unambiguous_JJ no_UH ._. 
It_PPH1 is_VBZ always_RR laborious_JJ to_TO find_VVI the_AT components_NN2 of_IO a_AT1 vector_NN1 ,_, so_CS we_PPIS2 are_VBR not_XX much_RR better_JJR off_RP with_IW A_ZZ1 than_CSN with_IW H_ZZ1 or_CC B._NP1 It_PPH1 turns_VVZ out_RP however_RR that_CST A_ZZ1 is_VBZ a_AT1 more_RGR basic_JJ quantity_NN1 of_IO physics_NN1 than_CSN B._NP1 Since_CS B_ZZ1 is_VBZ given_VVN by_II the_AT curl_NN1 of_IO A_ZZ1 it_PPH1 is_VBZ possible_JJ that_CST A_ZZ1 is_VBZ finite_JJ while_CS its_APPGE curl_NN1 is_VBZ zero_MC ._. 
Interestingly_RR ,_, under_II these_DD2 conditions_NN2 A_ZZ1 has_VHZ an_AT1 effect_NN1 on_II certain_JJ quantum-mechanical_JJ phenomena_NN2 It_PPH1 would_VM be_VBI unfair_JJ both_RR to_II the_AT vector_NN1 potential_NN1 and_CC to_II quantum_NN1 mechanics_NN2 to_TO say_VVI that_CST none_PN of_IO those_DD2 formulations_NN2 have_VH0 engineering_NN1 applications_NN2 (_( in_II fact_NN1 the_AT most_RGT sensitive_JJ magnetometer_NN1 built_VVN to_II date_NN1 is_VBZ based_VVN on_II that_DD1 kind_NN1 of_IO theory_NN1 )_) ,_, but_CCB by_RR31 and_RR32 large_RR33 engineers_NN2 would_VM n't_XX lose_VVI much_DA1 sleep_NN1 if_CS the_AT use_NN1 of_IO A_ZZ1 were_VBDR banned_VVN with_IW immediate_JJ effect_NN1 ._. 
The_AT vector_NN1 potential_NN1 is_VBZ not_XX a_AT1 popular_JJ variable_NN1 among_II engineers_NN2 ,_, maybe_RR because_CS it_PPH1 is_VBZ not_XX directly_RR measurable_JJ ._. 
There_EX are_VBR no_AT instruments_NN2 capable_JJ to_TO measure_VVI the_AT magnitude_NN1 or_CC direction_NN1 of_IO A._NNU In_II my_APPGE opinion_NN1 A_ZZ1 is_VBZ a_AT1 useful_JJ thing_NN1 if_CS used_VVN with_IW moderation_NN1 ._. 
It_PPH1 helped_VVD us_PPIO2 to_TO derive_VVI Biot-Savart_NP1 's_GE law_NN1 ,_, it_PPH1 will_VM come_VVI handy_JJ later_JJR in_II solving_VVG certain_JJ radiation_NN1 problems_NN2 ,_, and_CC it_PPH1 often_RR leads_VVZ to_II nice_JJ formulae_NN2 ,_, e.g._REX for_IF the_AT magnetic_JJ flux_NN1 crossing_VVG a_AT1 surface_NN1 ,_, defined_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) that_CST may_VM be_VBI rewritten_VVN in_II31 terms_II32 of_II33 the_AT vector_NN1 potential_NN1 as_CSA follows_VVZ :_: &lsqb;_( formula_NN1 &rsqb;_) where_RRQ C_ZZ1 is_VBZ the_AT curve_NN1 enclosing_VVG the_AT surface_NN1 ._. 
We_PPIS2 have_VH0 now_RT collected_VVN a_AT1 good_JJ number_NN1 of_IO formulae_NN2 which_DDQ will_VM serve_VVI us_PPIO2 well_RR in_II the_AT following_JJ sections_NN2 ._. 
Before_II going_VVG on_RP ,_, just_RR a_AT1 few_DA2 words_NN2 about_II the_AT classification_NN1 of_IO steady_JJ currents_NN2 ._. 
It_PPH1 may_VM be_VBI roughly_RR divided_VVN into_II two_MC parts_NN2 :_: magnetostatics_NN2 and_CC the_AT rest_NN1 ._. 
We_PPIS2 shall_VM discuss_VVI the_AT rest_NN1 first_MD (_( Sections_NN2 3.23.5_MC )_) ,_, and_CC that_DD1 will_VM give_VVI us_PPIO2 some_DD idea_NN1 of_IO the_AT relative_JJ significance_NN1 of_IO electric_JJ and_CC magnetic_JJ fields_NN2 ._. 
Sections_NN2 3.63.16_MC are_VBR concerned_JJ with_IW magnetostatics_NN2 ,_, where_CS electric_JJ fields_NN2 are_VBR assumed_VVN to_TO be_VBI zero_MC and_CC the_AT interrelationship_NN1 of_IO J_ZZ1 ,_, H_ZZ1 ,_, and_CC B_ZZ1 are_VBR studied_VVN ._. 
Since_CS we_PPIS2 have_VH0 so_RG many_DA2 variables_NN2 ,_, and_CC since_CS in_II each_DD1 physical_JJ configuration_NN1 only_RR some_DD of_IO them_PPHO2 appear_VV0 ,_, we_PPIS2 will_VM record_VVI (_( just_RR for_IF this_DD1 chapter_NN1 )_) the_AT non-zero_JJ variables_NN2 in_II each_DD1 section_NN1 heading._NNU 3.2_MC ._. 
The_AT defocusing_NN1 of_IO an_AT1 electron_NN1 beam_NN1 (_( J_ZZ1 ,_, E_ZZ1 ,_, D_ZZ1 ,_, p_ZZ1 ,_, H_ZZ1 ,_, B_ZZ1 )_) We_PPIS2 have_VH0 so_RG far_RR talked_VVN about_II positive_JJ and_CC negative_JJ charges_NN2 ,_, about_II point_NN1 charges_NN2 ,_, and_CC distributed_JJ charge_NN1 ._. 
I_PPIS1 shall_VM now_RT introduce_VVI the_AT concept_NN1 of_IO an_AT1 elementary_JJ charge_NN1 ,_, 1.6_MC 10_MC -19_MC C_NP1 ,_, carried_VVN by_II an_AT1 elementary_JJ particle_NN1 called_VVN the_AT electron_NN1 ._. 
This_DD1 is_VBZ not_XX a_AT1 necessity_NN1 ._. 
It_PPH1 is_VBZ possible_JJ to_TO study_VVI electromagnetic_JJ theory_NN1 without_IW ever_RR mentioning_VVG the_AT word_NN1 electron_NN1 ,_, but_CCB since_CS it_PPH1 has_VHZ become_VVN such_DA a_AT1 household_NN1 word_NN1 and_CC is_VBZ used_VVN so_RG often_RR we_PPIS2 can_VM just_RR as_RR21 well_RR22 make_VVI use_NN1 of_IO it_PPH1 ._. 
In_II the_AT present_JJ section_NN1 we_PPIS2 shall_VM consider_VVI a_AT1 cylindrical_JJ electron_NN1 beam_NN1 of_IO radius_NN1 a_AT1 in_II which_DDQ the_AT charge_NN1 density_NN1 is_VBZ uniform_JJ (_( p_ZZ1 =_FO Po_NP1 )_) and_CC all_DB electrons_NN2 travel_VV0 with_IW velocity_NN1 v._II We_PPIS2 are_VBR not_XX going_VVGK to_TO enquire_VVI into_II the_AT details_NN2 how_RRQ such_DA beams_NN2 can_VM be_VBI produced_VVN (_( it_PPH1 belongs_VVZ to_II the_AT subject_NN1 of_IO physical_JJ electronics_NN1 )_) ;_; we_PPIS2 shall_VM accept_VVI the_AT fact_NN1 that_CST the_AT beam_NN1 exists_VVZ and_CC will_VM try_VVI to_TO work_VVI out_RP the_AT forces_NN2 on_II the_AT outermost_JJ electrons_NN2 ._. 
There_EX is_VBZ now_RT cylindrical_JJ symmetry_NN1 and_CC a_AT1 net_JJ charge_NN1 per_II unit_NN1 length_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, yielding_VVG for_IF the_AT radial_JJ component_NN1 of_IO the_AT electric_JJ field_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT magnetic_JJ field_NN1 may_VM be_VBI obtained_VVN from_II Ampre_NP1 's_GE law_NN1 (_( eqn_NN1 (_( 3.21_MC )_) )_) as_CSA follows_VVZ &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS the_AT line_NN1 integral_JJ is_VBZ taken_VVN over_II the_AT circle_NN1 of_IO radius_NN1 a_AT1 (_( Fig._NN1 3.2_MC )_) ,_, I_ZZ1 is_VBZ the_AT total_JJ current_NN1 of_IO the_AT beam_NN1 and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT azimuthal_JJ component_NN1 of_IO the_AT magnetic_JJ field_NN1 in_II the_AT cylindrical_JJ coordinate_NN1 system_NN1 specified_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
From_II eqn_NN1 (_( 3.26_MC )_) we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT electric_JJ force_NN1 on_II an_AT1 electron_NN1 of_IO charge_NN1 e_ZZ1 travelling_VVG at_II the_AT edge_NN1 of_IO the_AT beam_NN1 (_( R_ZZ1 =_FO a_ZZ1 )_) is_VBZ it_PPH1 is_VBZ in_II the_AT radial_JJ direction_NN1 pointing_VVG outwards_RL ._. 
The_AT magnetic_JJ force_NN1 is_VBZ perpendicular_JJ both_RR to_II the_AT direction_NN1 of_IO motion_NN1 (_( +z_FO axis_NN1 )_) and_CC the_AT direction_NN1 of_IO magnetic_JJ field_NN1 (_( &lsqb;_( formula_NN1 &rsqb;_) direction_NN1 )_) ._. 
The_AT vectorial_JJ product_NN1 v_ZZ1 X_ZZ1 B_ZZ1 gives_VVZ an_AT1 inward_JJ force_NN1 in_II the_AT radial_JJ direction_NN1 ._. 
Hence_RR the_AT net_JJ force_NN1 on_II the_AT electron_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Noting_VVG that_CST &lsqb;_( formula_NN1 &rsqb;_) and_CC anticipating_VVG the_AT relationship_NN1 &lsqb;_( formula_NN1 &rsqb;_) to_TO be_VBI derived_VVN in_II Section_NN1 5.7_MC ,_, we_PPIS2 get_VV0 the_AT following_JJ formula_NN1 for_IF the_AT force_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS c_ZZ1 is_VBZ the_AT velocity_NN1 of_IO light_NN1 ._. 
It_PPH1 may_VM be_VBI seen_VVN from_II the_AT above_JJ equation_NN1 that_CST the_AT magnetic_JJ force_NN1 is_VBZ negligible_JJ in_II31 comparison_II32 with_II33 the_AT electric_JJ force_NN1 unless_CS the_AT velocity_NN1 of_IO the_AT electron_NN1 approaches_VVZ the_AT velocity_NN1 of_IO light_NN1 ._. 
Thus_RR the_AT conclusion_NN1 is_VBZ that_CST ,_, owing_II21 to_II22 the_AT repulsive_JJ forces_NN2 between_II the_AT electrons_NN2 ,_, a_AT1 cylindrical_JJ electron_NN1 beam_NN1 is_VBZ unstable_JJ ._. 
Using_VVG a_AT1 technical_JJ term_NN1 borrowed_VVN from_II electron_NN1 optics_NN1 we_PPIS2 could_VM say_VVI that_CST the_AT electron_NN1 beam_NN1 gets_VVZ defocused_JJ ._. 
The_AT other_JJ conclusion_NN1 (_( as_CS31 far_CS32 as_CS33 it_PPH1 is_VBZ permissible_JJ to_TO generalize_VVI from_II a_AT1 single_JJ example_NN1 )_) is_VBZ more_RGR important_JJ in_II principle_NN1 ._. 
It_PPH1 is_VBZ concerned_JJ with_IW the_AT relative_JJ magnitudes_NN2 of_IO electric_JJ and_CC magnetic_JJ forces_NN2 ._. 
Since_CS charged_JJ particles_NN2 rarely_RR travel_VV0 close_RR to_II the_AT velocity_NN1 of_IO light_NN1 we_PPIS2 may_VM conclude_VVI that_CST the_AT magnetic_JJ forces_NN2 are_VBR by_II orders_NN2 of_IO magnitude_NN1 smaller_JJR than_CSN the_AT electric_JJ forces_NN2 ._. 
Why_RRQ is_VBZ it_PPH1 then_RT that_CST we_PPIS2 have_VH0 no_AT difficulties_NN2 in_II practice_NN1 in_II observing_VVG magnetic_JJ fields_NN2 ?_? 
The_AT reason_NN1 is_VBZ ,_, of_RR21 course_RR22 ,_, that_CST there_EX are_VBR two_MC kinds_NN2 of_IO electric_JJ charges_NN2 and_CC their_APPGE effect_NN1 usually_RR cancels_VVZ ,_, so_CS the_AT small_JJ magnetic_JJ field_NN1 has_VHZ a_AT1 chance_NN1 of_IO getting_VVG observed_VVN ._. 
And_CC that_DD1 brings_VVZ us_PPIO2 to_II the_AT ext_NN1 example._NNU 3.3_MC Pinch_NN1 effect_NN1 (_( J_ZZ1 ,_, H_ZZ1 ,_, B_ZZ1 )_) In_II contrast_NN1 to_II our_APPGE previous_JJ example_NN1 we_PPIS2 shall_VM now_RT investigate_VVI a_AT1 cylindrical_JJ beam_NN1 consisting_VVG of_IO two_MC kinds_NN2 of_IO charge_NN1 carriers_NN2 :_: negative_JJ electrons_NN2 and_CC some_DD positive_JJ particles_NN2 ,_, which_DDQ I_PPIS1 do_VD0 not_XX wish_VVI to_TO be_VBI more_RGR precise_JJ about_II at_II the_AT moment_NN1 ._. 
We_PPIS2 shall_VM assume_VVI that_CST the_AT two_MC kinds_NN2 of_IO particles_NN2 have_VH0 equal_JJ densities_NN2 and_CC move_VV0 in_II opposite_JJ directions_NN2 ._. 
As_II a_AT1 result_NN1 there_EX is_VBZ no_AT net_JJ space_NN1 charge_NN1 and_CC hence_RR no_AT electric_JJ field_NN1 ._. 
The_AT currents_NN2 ,_, on_II the_AT other_JJ hand_NN1 ,_, do_VD0 not_XX cancel_VVI because_CS they_PPHS2 represent_VV0 opposite_JJ charges_NN2 moving_VVG in_II opposite_JJ directions_NN2 (_( remember_VV0 ,_, minus_II one_PN1 times_VVZ minus_II one_PN1 is_VBZ equal_JJ to_II plus_II one_MC1 )_) ._. 
The_AT magnetic_JJ field_NN1 strength_NN1 is_VBZ given_VVN again_RT by_II eqn_NN1 (_( 3.27_MC )_) ;_; we_PPIS2 only_RR need_VV0 to_TO substitute_VVI for_IF the_AT current_JJ In_II +_FO Ip_VV0 where_RRQ the_AT subscripts_NN2 n_ZZ1 and_CC p_ZZ1 refer_VV0 to_II negative_JJ and_CC positive_JJ particles_NN2 respectively_RR ._. 
Hence_RR the_AT magnetic_JJ force_NN1 on_II the_AT electron_NN1 at_II the_AT edge_NN1 of_IO the_AT beam_NN1 (_( at_II R_ZZ1 =_FO a_ZZ1 )_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC on_II the_AT positive_JJ particle_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Note_VV0 that_CST the_AT force_NN1 is_VBZ inwards_RL ,_, and_CC in_II the_AT same_DA direction_NN1 for_IF both_DB2 particles_NN2 ._. 
Let_VV0 us_PPIO2 distinguish_VVI two_MC cases_NN2 ._. 
Both_DB2 particles_NN2 are_VBR mobile_JJ In_II that_DD1 case_NN1 both_RR of_IO them_PPHO2 will_VM move_VVI inwards_RL under_II the_AT effect_NN1 of_IO magnetic_JJ force_NN1 ._. 
But_CCB if_CS the_AT diameter_NN1 of_IO the_AT beam_NN1 is_VBZ reduced_VVN ,_, eqns_NN2 (_( 3.33_MC )_) and_CC (_( 3.34_MC )_) tell_VV0 us_PPIO2 that_CST the_AT forces_NN2 are_VBR even_RR larger_JJR ._. 
What_DDQ happens_VVZ then_RT ?_? 
If_CS everything_PN1 was_VBDZ uniform_JJ then_RT the_AT beam_NN1 diameter_NN1 would_VM go_VVI on_RP decreasing_VVG ._. 
In_II practice_NN1 however_RRQV the_AT beam_NN1 is_VBZ not_XX uniform_JJ and_CC does_VDZ not_XX possess_VVI perfect_JJ cylindrical_JJ symmetry_NN1 ._. 
Under_II these_DD2 conditions_NN2 the_AT motion_NN1 of_IO the_AT particles_NN2 is_VBZ fairly_RR complicated_JJ ,_, and_CC of_RR21 course_RR22 our_APPGE model_NN1 is_VBZ unable_JK to_TO predict_VVI the_AT detailed_JJ behaviour_NN1 of_IO the_AT particles_NN2 ._. 
Nevertheless_RR a_AT1 few_DA2 qualitative_JJ conclusions_NN2 may_VM be_VBI drawn_VVN without_RR doing_VDG any_DD further_JJR mathematics_NN1 ._. 
If_CS the_AT cross-section_NN1 happens_VVZ to_TO be_VBI smaller_JJR at_II a_AT1 certain_JJ place_NN1 ,_, then_RT the_AT forces_NN2 are_VBR larger_JJR there_RL than_CSN at_II the_AT neighbouring_JJ cross-sections_NN2 ,_, so_CS the_AT beam_NN1 will_VM be_VBI further_RRR constricted_VVN ,_, etc._RA ,_, leading_VVG to_II the_AT so-called_JJ sausage_NN1 instability_NN1 (_( Fig._NN1 3.3(a)_FO )_) ._. 
If_CS the_AT magnetic_JJ field_NN1 happens_VVZ to_TO be_VBI larger_JJR at_II one_MC1 side_NN1 than_CSN at_II the_AT other_JJ side_NN1 ,_, then_RT the_AT beam_NN1 will_VM be_VBI deflected_VVN towards_II the_AT weaker_JJR field_NN1 which_DDQ makes_VVZ the_AT field_NN1 even_RR weaker_JJR ,_, etc._RA ,_, leading_VVG to_II the_AT so-called_JJ kink_NN1 instability_NN1 (_( Fig._NN1 3.3(b)_FO )_) ._. 
The_AT positive_JJ particle_NN1 is_VBZ immobile_JJ In_II practice_NN1 this_DD1 means_VVZ that_CST the_AT positive_JJ ions_NN2 are_VBR part_NN1 of_IO the_AT crystal_NN1 lattice_NN1 ._. 
There_EX is_VBZ then_RT no_AT magnetic_JJ force_NN1 on_II the_AT ions_NN2 ,_, only_RR on_II the_AT electrons_NN2 ._. 
So_RR the_AT electrons_NN2 want_VV0 to_TO move_VVI inwards_RL but_CCB can_VM not_XX because_CS the_AT ions_NN2 hold_VV0 them_PPHO2 back_RP by_II virtue_NN1 of_IO their_APPGE electrostatic_JJ attraction_NN1 ._. 
Naturally_RR ,_, if_CS the_AT ions_NN2 attract_VV0 the_AT electrons_NN2 the_AT converse_NN1 is_VBZ true_JJ as_RR21 well_RR22 ,_, i.e._REX owing_II21 to_II22 the_AT magnetic_JJ force_NN1 on_II the_AT electrons_NN2 ,_, there_EX is_VBZ also_RR an_AT1 inward_JJ force_NN1 upon_II the_AT ions_NN2 ._. 
So_RR the_AT whole_JJ crystal_NN1 structure_NN1 tries_VVZ to_TO contract_VVI or_CC in_II other_JJ words_NN2 the_AT material_NN1 is_VBZ under_II pressure_NN1 (_( see_VV0 Example_NN1 3.2_MC )_) ._. 
In_II any_DD practical_JJ situation_NN1 this_DD1 pressure_NN1 is_VBZ small_JJ and_CC can_VM be_VBI neglected_VVN when_CS calculating_VVG the_AT forces_NN2 on_II current-carrying_JJ conductors._NNU 3.4_MC ._. 
Flow_NN1 patterns_NN2 and_CC Ohm_NP1 's_GE law_NN1 (_( J_ZZ1 ,_, E_ZZ1 ,_, D_ZZ1 )_) Up_II21 to_II22 now_RT we_PPIS2 have_VH0 not_XX enquired_VVN into_II the_AT question_NN1 of_IO how_RRQ the_AT current_NN1 arose_VVD ._. 
We_PPIS2 just_RR assumed_VVD that_CST certain_JJ charge_NN1 carriers_NN2 moved_VVN with_IW certain_JJ velocities_NN2 ._. 
Now_RT we_PPIS2 shall_VM say_VVI that_CST in_II a_AT1 class_NN1 of_IO materials_NN2 the_AT current_JJ density_NN1 at_II any_DD point_NN1 is_VBZ proportional_JJ to_II the_AT electric_JJ field_NN1 ;_; in_II mathematical_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT conductivity_NN1 of_IO the_AT material_NN1 ._. 
The_AT above_JJ equation_NN1 is_VBZ spite_NN1 of_IO its_APPGE utmost_JJ simplicity_NN1 represents_VVZ an_AT1 extremely_RR good_JJ guess_NN1 ._. 
It_PPH1 is_VBZ valid_JJ for_IF nearly_RR all_DB materials_NN2 up_II21 to_II22 quite_RG high_JJ electric_JJ fields_NN2 ._. 
Will_VM there_EX be_VBI a_AT1 magnetic_JJ field_NN1 ?_? 
Yes_UH ,_, of_RR21 course_RR22 ,_, whenever_RRQV there_EX is_VBZ a_AT1 current_JJ there_EX is_VBZ a_AT1 magnetic_JJ field_NN1 as_RR21 well_RR22 ._. 
But_CCB the_AT effect_NN1 of_IO the_AT magnetic_JJ field_NN1 upon_II the_AT motion_NN1 of_IO the_AT electrons_NN2 is_VBZ nearly_RR always_RR negligible_JJ (_( one_MC1 exception_NN1 causing_VVG the_AT pinch_NN1 effect_NN1 is_VBZ mentioned_VVN in_II the_AT previous_JJ section_NN1 )_) ._. 
Thus_RR we_PPIS2 may_VM safely_RR ignore_VVI the_AT magnetic_JJ field_NN1 when_CS determining_VVG the_AT lines_NN2 of_IO current_JJ flow_NN1 ._. 
We_PPIS2 are_VBR then_RT left_VVN with_IW only_RR one_MC1 equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) ,_, depending_II21 on_II22 our_APPGE preference_NN1 ._. 
Eqn_NN1 (_( 3.38_MC )_) tells_VVZ us_PPIO2 that_CST there_EX is_VBZ some_DD analogy_NN1 (_( see_VV0 Example_NN1 3.3_MC )_) with_IW the_AT electrostatic_JJ case_NN1 treated_VVN in_II Chapter_NN1 2_MC ._. 
We_PPIS2 may_VM ,_, in_II fact_NN1 ,_, reinterpret_VV0 any_DD of_IO the_AT diagrams_NN2 of_IO Figs_NN2 (_( 2.25_MC )_) -(2.31)_NNU by_II assuming_VVG that_CST the_AT whole_JJ space_NN1 is_VBZ filled_VVN with_IW a_AT1 material_NN1 of_IO conductivity_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT field_NN1 lines_NN2 are_VBR now_RT the_AT lines_NN2 of_IO current_JJ flow_NN1 as_RR21 well_RR22 ._. 
Is_VBZ the_AT analogy_NN1 perfect_JJ ?_? 
Not_XX really_RR ,_, because_CS (_( 1_MC1 )_) as_CSA current_JJ flows_NN2 there_EX is_VBZ some_DD potential_JJ drop_NN1 in_II the_AT electrode_NN1 itself_PPX1 ,_, (_( 2_MC )_) zero_MC conductivity_NN1 for_IF part_NN1 of_IO the_AT space_NN1 can_VM not_XX be_VBI electrostatically_RR modelled_VVN since_CS there_EX are_VBR no_AT dielectrics_NN1 with_IW Er_NP1 =_FO 0_MC ,_, and_CC (_( 3_MC )_) when_RRQ current_JJ flows_NN2 through_II two_MC materials_NN2 of_IO different_JJ conductivity_NN1 there_EX is_VBZ generally_RR a_AT1 surface_NN1 charge_NN1 at_II the_AT boundary_NN1 (_( see_VV0 Example_NN1 3.4_MC )_) ._. 
It_PPH1 is_VBZ not_XX worth_II discussing_VVG any_DD of_IO these_DD2 complications_NN2 because_CS one_PN1 is_VBZ rarely_RR called_VVN upon_II to_TO work_VVI out_RP lines_NN2 of_IO current_JJ in_II a_AT1 conductor_NN1 ._. 
We_PPIS2 shall_VM rather_RR return_VVI to_II a_AT1 very_RG simple_JJ geometrical_JJ configuration_NN1 for_IF deriving_VVG Ohm_NP1 's_GE law_NN1 ._. 
We_PPIS2 shall_VM take_VVI a_AT1 piece_NN1 of_IO cylindrical_JJ material_NN1 of_IO length_NN1 l_ZZ1 and_CC cross-section_NN1 S_ZZ1 and_CC apply_VV0 a_AT1 voltage_NN1 between_II the_AT ends_NN2 ._. 
The_AT electric_JJ field_NN1 is_VBZ then_RT &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT current_JJ density_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC the_AT current_JJ &lsqb;_( formula_NN1 &rsqb;_) ._. 
From_II circuit_NN1 theory_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS R_ZZ1 is_VBZ the_AT resistance_NN1 of_IO the_AT material_NN1 ._. 
Comparing_VVG eqns_NN2 (_( 3.41_MC )_) and_CC (_( 3.42_MC )_) we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Since_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ by_II definition_NN1 the_AT resistivity_NN1 we_PPIS2 get_VV0 the_AT result_NN1 you_PPY learned_VVD in_II school_NN1 that_CST the_AT electrical_JJ resistance_NN1 is_VBZ proportional_JJ to_II the_AT resistivity_NN1 and_CC the_AT length_NN1 of_IO the_AT sample_NN1 and_CC inversely_RR proportional_JJ to_II its_APPGE cross-section_NN1 ._. 
A_AT1 more_RGR general_JJ definition_NN1 of_IO resistance_NN1 valid_JJ for_IF varying_JJ cross-sections_NN2 may_VM be_VBI easily_RR arrived_VVN at_II ,_, but_CCB it_PPH1 is_VBZ hardly_RR worth_II the_AT trouble_NN1 ._. 
The_AT essential_JJ thing_NN1 is_VBZ that_CST the_AT relation_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ equivalent_JJ to_II Ohm_NP1 's_GE law_NN1 ._. 
Risking_VVG the_AT dismay_NN1 of_IO circuit_NN1 engineers_NN2 some_DD theoreticians_NN2 do_VD0 ,_, in_II fact_NN1 ,_, refer_VV0 to_II eqn_NN1 (_( 3.35_MC )_) as_CSA Ohm_NP1 's_GE law._NNU 3.5_MC ._. 
Electron_NN1 flow_NN1 between_II parallel_JJ plates_NN2 &lsqb;_( J_ZZ1 ,_, p_ZZ1 ,_, E_ZZ1 ,_, D_ZZ1 )_) We_PPIS2 shall_VM investigate_VVI here_RL just_RR one_PN1 more_RGR physical_JJ configuration_NN1 where_CS in_II31 spite_II32 of_II33 having_VHG a_AT1 finite_JJ current_JJ the_AT magnetic_JJ field_NN1 may_VM be_VBI disregarded_VVN ._. 
The_AT new_JJ feature_NN1 will_VM be_VBI a_AT1 spatially_RR varying_VVG space-charge_JJ density_NN1 ._. 
Let_VV0 us_PPIO2 take_VVI two_MC parallel_RR conducting_VVG plates_NN2 in_II a_AT1 vacuum_NN1 ,_, one_MC1 of_IO them_PPHO2 endowed_VVN with_IW the_AT property_NN1 that_CST it_PPH1 is_VBZ capable_JJ of_IO emitting_VVG electrons_NN2 ._. 
We_PPIS2 shall_VM further_RRR apply_VVI a_AT1 voltage_NN1 between_II the_AT plates_NN2 so_CS21 that_CS22 the_AT electrons_NN2 emitted_VVD by_II electrode_NN1 1_MC1 are_VBR attracted_VVN to_TO electrode_VVI 2_MC (_( Fig._NN1 3.4_MC )_) ._. 
The_AT aim_NN1 is_VBZ to_TO calculate_VVI the_AT potential_JJ distribution_NN1 between_II the_AT plates_NN2 ,_, and_CC the_AT relationship_NN1 between_II applied_JJ voltage_NN1 and_CC the_AT magnitude_NN1 of_IO resulting_JJ current_JJ ._. 
We_PPIS2 can_VM reduce_VVI the_AT problem_NN1 to_II a_AT1 one-dimensional_JJ one_PN1 by_II claiming_VVG that_CST the_AT plates_NN2 are_VBR infinitely_RR large_JJ ._. 
Alternatively_RR ,_, we_PPIS2 may_VM say_VVI that_CST the_AT plates_NN2 are_VBR so_RG close_JJ to_II each_PPX221 other_PPX222 (_( as_CSA it_PPH1 would_VM be_VBI in_II a_AT1 practical_JJ diode_NN1 )_) that_CST the_AT electron_NN1 beam_NN1 has_VHZ n't_XX got_VVN a_AT1 chance_NN1 to_TO spread_VVI ._. 
Whichever_DDQV way_NN1 we_PPIS2 look_VV0 at_II it_PPH1 we_PPIS2 can_VM disregard_VVI with_IW clear_JJ conscience_NN1 both_RR the_AT radial_JJ electric_JJ field_NN1 and_CC the_AT magnetic_JJ field_NN1 ._. 
Hence_RR the_AT variables_NN2 of_IO interest_NN1 are_VBR the_AT current_JJ density_NN1 ,_, the_AT electron_NN1 velocity_NN1 ,_, the_AT space-charge_JJ density_NN1 ,_, the_AT longitudinal_JJ electric_JJ field_NN1 strength_NN1 ,_, and_CC the_AT potential_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 can_VM arbitrarily_RR assign_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS V_ZZ1 is_VBZ the_AT potential_JJ difference_NN1 applied_VVD ._. 
The_AT electron_NN1 velocity_NN1 at_II the_AT point_NN1 z_ZZ1 may_VM be_VBI worked_VVN out_RP by_II the_AT simple_JJ consideration_NN1 that_CST the_AT electron_NN1 gains_VVZ kinetic_JJ energy_NN1 at_II the_AT expense_NN1 of_IO its_APPGE potential_JJ energy_NN1 (_( the_AT same_DA idea_NN1 as_CSA in_II mechanics_NN2 )_) ._. 
Hence_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
Assuming_VVG now_CS21 that_CS22 the_AT electrons_NN2 are_VBR emitted_VVN with_IW zero_MC initial_JJ velocity_NN1 ,_, v(0)_FO =_FO 0_MC ,_, we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Note_VV0 that_CST e_ZZ1 is_VBZ negative_JJ and_CC that_DD1 ,_, for_IF the_AT above_JJ equation_NN1 to_TO apply_VVI ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT charge_NN1 density_NN1 may_VM be_VBI obtained_VVN from_II the_AT relationship_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS J_ZZ1 is_VBZ a_AT1 positive_JJ constant._NNU p_ZZ1 must_VM of_RR21 course_RR22 be_VBI negative_JJ because_CS it_PPH1 represents_VVZ the_AT space-charge_JJ density_NN1 of_IO electrons_NN2 ._. 
In_II this_DD1 example_NN1 the_AT charges_NN2 are_VBR in_II motion_NN1 and_CC the_AT charge_NN1 density_NN1 varies_VVZ from_II point_NN1 to_II point_NN1 but_CCB the_AT density_NN1 at_II a_AT1 given_JJ point_NN1 z_ZZ1 is_VBZ not_XX dependent_JJ on_II time_NNT1 ._. 
Hence_RR we_PPIS2 are_VBR faced_VVN here_RL with_IW an_AT1 electrostatic_JJ problem_NN1 which_DDQ may_VM be_VBI solved_VVN with_IW the_AT aid_NN1 of_IO Poisson_NP1 's_GE equation_NN1 (_( eqn_NN1 (_( 2.12_MC )_) )_) ._. 
Substituting_VVG into_II it_PPH1 the_AT value_NN1 of_IO p_ZZ1 from_II eqn_NN1 (_( 3.46_MC )_) we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ a_AT1 reasonable-looking_JJ differential_JJ equation_NN1 ;_; I_PPIS1 leave_VV0 the_AT solution_NN1 to_II you_PPY (_( Example_NN1 3.5_MC )_) ._. 
The_AT main_JJ reason_NN1 for_IF showing_VVG this_DD1 example_NN1 is_VBZ not_XX its_APPGE intrinsic_JJ value_NN1 to_II applied_JJ scientists_NN2 (_( the_AT problem_NN1 of_IO space-charged-limited_JJ diodes_NN2 is_VBZ no_RR21 longer_RR22 in_II the_AT forefront_NN1 of_IO interest_NN1 )_) but_CCB to_TO demonstrate_VVI the_AT applicability_NN1 of_IO Poisson_NP1 's_GE equation_NN1 under_II conditions_NN2 of_IO steady_JJ current_JJ flow._NNU 3.6_MC ._. 
The_AT magnetic_JJ field_NN1 due_JJ to_TO line_VVI currents_NN2 (_( J_ZZ1 ,_, H_ZZ1 ,_, B_ZZ1 )_) Assume_VV0 that_CST a_AT1 current_JJ I_ZZ1 flows_VVZ along_II the_AT z_ZZ1 axis_NN1 from_II minus_NN1 infinity_NN1 to_II plus_NN1 infinity_NN1 (_( Fig._NN1 3.5(a)_FO )_) and_CC find_VV0 the_AT magnetic_JJ field_NN1 at_II a_AT1 distance_NN1 R_ZZ1 from_II the_AT current_JJ ._. 
Owing_II21 to_II22 axial_JJ symmetry_NN1 the_AT magnetic_JJ field_NN1 must_VM be_VBI constant_JJ at_II a_AT1 radius_NN1 R_ZZ1 ,_, hence_RR the_AT application_NN1 of_IO Ampre_NP1 's_GE law_NN1 yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, or_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
In_II fact_NN1 we_PPIS2 have_VH0 derived_VVN this_DD1 relationship_NN1 in_II Section_NN1 3.2_MC ._. 
It_PPH1 makes_VVZ no_AT difference_NN1 whether_CSW the_AT current_NN1 is_VBZ distributed_VVN within_II a_AT1 radius_NN1 a_AT1 or_CC concentrated_VVN at_II the_AT axis_NN1 (_( only_RR the_AT enclosed_JJ current_JJ counts_NN2 )_) ._. 
Could_VM we_PPIS2 get_VVI the_AT same_DA result_NN1 by_II using_VVG our_APPGE formula_NN1 for_IF the_AT vector_NN1 potential_NN1 (_( eqn(3.17)_FO )_) ?_? 
We_PPIS2 could_VM ,_, although_CS we_PPIS2 would_VM run_VVI again_RT into_II the_AT problem_NN1 of_IO a_AT1 diverging_JJ integral_JJ owing_II21 to_II22 the_AT limits_NN2 at_II infinity_NN1 ._. 
We_PPIS2 spent_VVD quite_RG a_AT1 long_JJ time_NNT1 sorting_VVG out_RP this_DD1 problem_NN1 in_II the_AT electrostatic_JJ case_NN1 ,_, and_CC we_PPIS2 need_VM not_XX repeat_VVI the_AT argument_NN1 here_RL ._. 
Just_RR by_II analogy_NN1 with_IW eqn_NN1 (_( 2.45_MC )_) we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF obtaining_VVG B_ZZ1 we_PPIS2 take_VV0 the_AT curl_NN1 of_IO the_AT vector_NN1 potential_NN1 in_II a_AT1 cylindrical_JJ coordinate_NN1 system_NN1 &lsqb;_( formula_NN1 &rsqb;_) as_CSA follows_VVZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Recognizing_VVG that_CST nothing_PN1 can_VM change_VVI in_II the_AT &lsqb;_( formula_NN1 &rsqb;_) and_CC z_ZZ1 directions_NN2 (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ,_, the_AT only_JJ non-zero_JJ component_NN1 is_VBZ provided_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, in_II agreement_NN1 with_IW eqn_NN1 (_( 3.49_MC )_) For_IF two_MC line_NN1 currents_NN2 flowing_VVG in_II the_AT opposite_JJ directions_NN2 (_( Fig._NN1 3.5(b)_FO )_) ,_, we_PPIS2 may_VM write_VVI Ampre_NP1 's_GE law_NN1 twice_RR and_CC add_VVI the_AT magnetic_JJ fields_NN2 or_CC add_VV0 the_AT vector_NN1 potentials_NN2 ._. 
In_II either_DD1 case_NN1 we_PPIS2 obtain_VV0 &lsqb;_( formula_NN1 &rsqb;_) where_RRQ &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) are_VBR unit_NN1 vectors_NN2 in_II the_AT azimuthal_JJ directions_NN2 from_II the_AT two_MC line_NN1 currents_NN2 respectively._NNU 3.7_MC ._. 
The_AT magnetic_JJ field_NN1 due_II21 to_II22 a_AT1 ring_NN1 current_NN1 (_( J_ZZ1 ,_, H_ZZ1 ,_, B_ZZ1 )_) The_AT solution_NN1 for_IF line_NN1 currents_NN2 was_VBDZ simple_JJ enough_RR ._. 
Unfortunately_RR the_AT determination_NN1 of_IO the_AT magnetic_JJ field_NN1 for_IF a_AT1 ring_NN1 current_NN1 needs_VVZ a_AT1 lot_NN1 of_IO calculation_NN1 ._. 
We_PPIS2 shall_VM assume_VVI that_CST a_AT1 ring_NN1 of_IO radius_NN1 a_AT1 situated_JJ in_II the_AT z_ZZ1 =_FO 0_MC plane_NN1 carries_VVZ a_AT1 current_JJ I_ZZ1 (_( Fig._NN1 3.6_MC )_) and_CC we_PPIS2 wish_VV0 to_TO determine_VVI the_AT magnetic_JJ field_NN1 at_II the_AT point_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, i.e._REX at_II an_AT1 arbitrary_JJ point_NN1 in_II space_NN1 ._. 
We_PPIS2 shall_VM rely_VVI on_II the_AT vector-potential_JJ formulation_NN1 because_CS that_DD1 appears_VVZ to_TO be_VBI best_RRT suited_VVN for_IF utilizing_VVG the_AT axial_JJ symmetry_NN1 of_IO the_AT chosen_JJ geometry_NN1 ._. 
The_AT vector_NN1 potential_NN1 due_II21 to_II22 any_DD current_JJ element_NN1 is_VBZ always_RR in_II the_AT direction_NN1 of_IO the_AT current_JJ element_NN1 ._. 
Hence_RR if_CS the_AT current_NN1 is_VBZ everywhere_RL in_II the_AT azimuthal_JJ direction_NN1 the_AT vector_NN1 potential_NN1 must_VM be_VBI in_II the_AT same_DA direction_NN1 too_RR ._. 
Thus_RR even_RR before_II starting_VVG any_DD calculations_NN2 we_PPIS2 may_VM immediately_RR say_VVI that_CST A_ZZ1 will_VM only_RR have_VHI an_AT1 &lsqb;_( formula_NN1 &rsqb;_) component_NN1 and_CC that_DD1 will_VM be_VBI independent_JJ of_IO &lsqb;_( formula_NN1 &rsqb;_) ._. 
Taking_VVG the_AT coordinates_NN2 of_IO the_AT current_JJ elements_NN2 dl_MC as_CSA &lsqb;_( formula_NN1 &rsqb;_) (_( Fig._NN1 3.6_MC )_) and_CC noting_VVG that_CST &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 obtain_VV0 for_IF the_AT desired_JJ component_NN1 of_IO the_AT vector_NN1 potential_NN1 &lsqb;_( formula_NN1 &rsqb;_) or_CC &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS the_AT expression_NN1 for_IF r_ZZ1 has_VHZ been_VBN substituted_VVN ._. 
Since_CS &lsqb;_( formula_NN1 &rsqb;_) must_VM be_VBI independent_JJ of_IO &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 can_VM simplify_VVI eqn_NN1 (_( 3.56_MC )_) by_II taking_VVG &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT remaining_JJ integration_NN1 is_VBZ nonetheless_RR difficult_JJ ,_, and_CC not_XX expressible_JJ in_II31 terms_II32 of_II33 simple_JJ functions_NN2 ._. 
Mathematicians_NN2 ,_, wisely_RR foreseeing_VVG such_DA difficulties_NN2 ,_, worked_VVD out_RP the_AT theory_NN1 of_IO a_AT1 number_NN1 of_IO special_JJ functions_NN2 and_CC tabulated_VVD them_PPHO2 as_RR21 well_RR22 ._. 
In_II the_AT age_NN1 of_IO the_AT computer_NN1 their_APPGE work_NN1 is_VBZ no_RR21 longer_RR22 indispensable_JJ but_CCB is_VBZ still_RR useful_JJ as_II a_AT1 sort_NN1 of_IO short-hand_NN1 notation_NN1 ._. 
If_CS you_PPY look_VV0 up_RP the_AT relevant_JJ books_NN2 you_PPY will_VM find_VVI that_DD1 eqn_NN1 (_( 3.56_MC )_) may_VM be_VBI expressed_VVN in_II31 terms_II32 of_II33 elliptic_JJ integrals_NN2 ._. 
Having_VHG obtained_VVN the_AT vector_NN1 potential_NN1 the_AT magnetic_JJ field_NN1 may_VM be_VBI obtained_VVN by_II the_AT usual_JJ differentiation_NN1 which_DDQ ,_, incidentally_RR ,_, also_RR leads_VVZ to_II elliptic_JJ integrals_NN2 ._. 
There_EX is_VBZ nothing_PN1 fearful_JJ in_II elliptic_JJ integrals_NN2 Their_APPGE definitions_NN2 are_VBR fairly_RR simple_JJ ,_, they_PPHS2 are_VBR nicely_RR tabulated_VVN and_CC they_PPHS2 have_VH0 a_AT1 few_DA2 interesting_JJ interrelations_NN2 ._. 
All_DB that_DD1 is_VBZ easily_RR digestible_JJ ._. 
My_APPGE main_JJ reason_NN1 for_IF not_XX introducing_VVG elliptic_JJ integrals_NN2 here_RL (_( you_PPY can_VM though_RR attempt_VVI Examples_NN2 3.93.11_MC if_CS you_PPY wish_VV0 to_TO have_VHI some_DD experience_NN1 in_II handling_VVG them_PPHO2 )_) is_VBZ that_CST I_PPIS1 do_VD0 not_XX want_VVI to_TO burden_VVI your_APPGE memory_NN1 with_IW new_JJ formulae_NN2 ,_, and_CC besides_RR ,_, for_IF cases_NN2 of_IO most_DAT practical_JJ interest_NN1 (_( the_AT field_NN1 far_RR away_II21 from_II22 the_AT ring_NN1 and_CC in_II the_AT vicinity_NN1 of_IO the_AT axis_NN1 )_) eqn_NN1 (_( 3.56_MC )_) may_VM be_VBI integrated_VVN out_RP ,_, as_CSA will_VM be_VBI presently_RR seen_VVN ._. 
For_IF both_DB2 cases_NN2 mentioned_VVN above_RL (_( &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) )_) ,_, the_AT following_JJ inequality_NN1 is_VBZ valid_JJ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR we_PPIS2 can_VM expand_VVI the_AT denominator_NN1 of_IO the_AT integrand_NN1 as_CSA follows_VVZ :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
Substituting_VVG the_AT above_JJ approximate_JJ relation_NN1 into_II eqn_NN1 (_( 3.56_MC )_) we_PPIS2 can_VM perform_VVI the_AT integration_NN1 ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ._. 
Let_VV0 us_PPIO2 first_MD consider_VVI the_AT case_NN1 when_CS the_AT point_NN1 of_IO observation_NN1 is_VBZ far_RR away_II21 from_II22 the_AT ring_NN1 (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ;_; then_RT the_AT vector_NN1 potential_NN1 takes_VVZ the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ may_VM be_VBI rewritten_VVN in_II spherical_JJ coordinates_NN2 as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, whence_RRQ the_AT magnetic_JJ field_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Noting_VVG that_CST &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 get_VV0 for_IF the_AT components_NN2 of_IO the_AT magnetic_JJ field_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Can_VM you_PPY remember_VVI seeing_VVG these_DD2 self-same_DA components_NN2 somewhere_RL before_RT ?_? 
Well_RR ,_, the_AT constants_NN2 are_VBR different_JJ but_CCB apart_II21 from_II22 that_DD1 eqn_NN1 (_( 2.33_MC )_) ,_, the_AT formula_NN1 for_IF the_AT electric_JJ field_NN1 of_IO an_AT1 electric_JJ dipole_NN1 ,_, looks_VVZ the_AT same_DA ._. 
On_II the_AT basis_NN1 of_IO this_DD1 analogy_NN1 we_PPIS2 may_VM call_VVI a_AT1 ring_NN1 current_NN1 a_AT1 magnetic_JJ dipole_NN1 ,_, or_CC more_RGR precisely_RR we_PPIS2 should_VM say_VVI that_CST sufficiently_RR far_RR away_II21 from_II22 a_AT1 ring_NN1 current_NN1 the_AT magnetic_JJ field_NN1 appears_VVZ as_CS21 if_CS22 it_PPH1 was_VBDZ created_VVN by_II two_MC closely_RR spaced_VVN magnetic_JJ charges_NN2 (_( which_DDQ of_RR21 course_RR22 do_VD0 not_XX exist_VVI )_) ._. 
There_EX is_VBZ another_DD1 less_RGR obvious_JJ conclusion_NN1 at_II which_DDQ one_PN1 might_VM arrive_VVI by_II inspecting_VVG eqn_NN1 (_( 3.63_MC )_) ._. 
Notice_VV0 that_CST the_AT area_NN1 of_IO the_AT ring_NN1 &lsqb;_( formula_NN1 &rsqb;_) appears_VVZ as_CSA one_MC1 of_IO the_AT factors_NN2 in_II the_AT constant_JJ ._. 
It_PPH1 turns_VVZ out_RP (_( though_CS we_PPIS2 are_VBR not_XX going_VVGK to_TO prove_VVI it_PPH1 here_RL )_) that_CST the_AT exact_JJ shape_NN1 of_IO the_AT current_JJ loop_NN1 is_VBZ immaterial_JJ (_( not_XX unreasonable_JJ if_CS the_AT loop_NN1 is_VBZ far_RR away_RL )_) and_CC in_II the_AT general_JJ case_NN1 we_PPIS2 only_RR need_VV0 to_TO replace_VVI &lsqb;_( formula_NN1 &rsqb;_) by_II S_ZZ1 ,_, the_AT area_NN1 of_IO the_AT loop_NN1 ._. 
Turning_VVG now_RT to_II the_AT case_NN1 when_CS the_AT point_NN1 of_IO observation_NN1 is_VBZ in_II the_AT vicinity_NN1 of_IO the_AT axis_NN1 ,_, we_PPIS2 get_VV0 for_IF the_AT vector_NN1 potential_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC for_IF the_AT magnetic_JJ field_NN1 (_( in_II cylindrical_JJ coordinates_NN2 this_DD1 time_NNT1 )_) &lsqb;_( formula_NN1 &rsqb;_) ._. 
3.8_MC ._. 
The_AT magnetic_JJ field_NN1 inside_II a_AT1 solenoid_NN1 (_( J_ZZ1 ,_, H_ZZ1 ,_, B_ZZ1 )_) A_ZZ1 solenoid_NN1 is_VBZ a_AT1 tightly_RR wound_VVN coil_NN1 ._. 
Each_DD1 turn_NN1 may_VM be_VBI regarded_VVN equivalent_JJ to_II a_AT1 ring_NN1 in_II which_DDQ a_AT1 current_JJ I_ZZ1 flows_VVZ ._. 
For_IF calculating_VVG the_AT magnetic_JJ field_NN1 we_PPIS2 shall_VM take_VVI the_AT coordinate_NN1 system_NN1 shown_VVN in_II Fig._NN1 3.7_MC ,_, where_CS the_AT z_ZZ1 axis_NN1 coincides_VVZ with_IW the_AT axis_NN1 of_IO the_AT solenoid_NN1 ._. 
At_II an_AT1 arbitrary_JJ point_NN1 on_II the_AT axis_NN1 &lsqb;_( formula_NN1 &rsqb;_) the_AT magnetic_JJ field_NN1 may_VM be_VBI obtained_VVN by_II summing_VVG the_AT contribution_NN1 of_IO each_DD1 turn_NN1 ._. 
It_PPH1 is_VBZ actually_RR easier_JJR to_TO do_VDI the_AT calculation_NN1 if_CS instead_II21 of_II22 individual_NN1 turns_VVZ we_PPIS2 assume_VV0 that_CST the_AT current_NN1 is_VBZ continuously_RR distributed_VVN on_II the_AT surface_NN1 of_IO the_AT cylinder_NN1 ,_, and_CC work_VV0 in_II31 terms_II32 of_II33 &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT current_JJ per_II unit_NN1 length_NN1 ,_, obtained_VVN by_II dividing_VVG the_AT total_JJ current_JJ (_( NI_NP1 )_) by_II the_AT length_NN1 l_ZZ1 of_IO the_AT solenoid_NN1 ,_, i.e._REX &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT current_JJ flowing_JJ at_II z_ZZ1 in_II the_AT interval_NN1 dz_NNU is_VBZ then_RT &lsqb;_( formula_NN1 &rsqb;_) ,_, at_II a_AT1 distance_NN1 &lsqb;_( formula_NN1 &rsqb;_) from_II the_AT point_NN1 where_RRQ we_PPIS2 wish_VV0 to_TO determine_VVI the_AT magnetic_JJ field_NN1 ._. 
According_II21 to_II22 eqn_NN1 (_( 3.65_MC )_) such_DA a_AT1 current_NN1 will_VM create_VVI a_AT1 magnetic_JJ field_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT total_JJ magnetic_JJ field_NN1 may_VM be_VBI obtained_VVN by_II integrating_VVG over_II the_AT length_NN1 of_IO the_AT solenoid_NN1 as_CSA follows_VVZ :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
Changing_VVG the_AT integration_NN1 variable_NN1 to_II the_AT angle_NN1 &lsqb;_( formula_NN1 &rsqb;_) by_II the_AT relationship_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 may_VM perform_VVI the_AT integration_NN1 and_CC get_VVI at_II the_AT end_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS the_AT solenoid_NN1 is_VBZ very_RG long_JJ (_( la_FW )_) ,_, then_RT &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ of_RR21 course_RR22 not_XX valid_JJ near_II the_AT ends_NN2 but_CCB if_CS the_AT solenoid_NN1 is_VBZ long_RR enough_RR ,_, eqn_NN1 (_( 3.71_MC )_) may_VM be_VBI regarded_VVN valid_JJ for_IF most_DAT of_IO its_APPGE length_NN1 ._. 
Incidentally_RR ,_, eqn_NN1 (_( 3.71_MC )_) may_VM be_VBI derived_VVN much_RR more_RGR simply_RR if_CS we_PPIS2 make_VV0 the_AT a_JJ21 priori_JJ22 assumption_NN1 that_CST the_AT magnetic_JJ field_NN1 is_VBZ constant_JJ inside_II the_AT solenoid_NN1 and_CC zero_MC outside_RL ._. 
We_PPIS2 may_VM then_RT apply_VVI Ampre_NP1 's_GE law_NN1 to_II the_AT path_NN1 shown_VVN in_II Fig._NN1 3.8_MC yielding_VVG It_PPH1 is_VBZ a_AT1 bit_NN1 of_IO a_AT1 coincidence_NN1 that_CST eqns_NN2 (_( 3.71_MC )_) and_CC (_( 3.72_MC )_) agree_VV0 ._. 
The_AT reason_NN1 is_VBZ that_CST in_II the_AT latter_DA approach_NN1 the_AT magnetic_JJ field_NN1 is_VBZ overestimated_VVN near_II the_AT ends_NN2 inside_II the_AT solenoid_NN1 ,_, and_CC underestimated_VVN outside_II the_AT solenoid_NN1 ._. 
The_AT two_MC inaccuracies_NN2 just_RR21 about_RR22 balance_VV0 each_PPX221 other._PPX222 3.9_MC Further_JJR analogies_NN2 with_IW electrostatics_NN2 What_DDQ happens_VVZ in_II the_AT electrostatic_JJ case_NN1 if_CS we_PPIS2 fill_VV0 the_AT whole_JJ space_NN1 by_II a_AT1 dielectric_JJ ?_? 
The_AT electric_JJ field_NN1 remains_VVZ unchanged_JJ and_CC the_AT electric_JJ flux_NN1 density_NN1 increases_VVZ by_II a_AT1 factor_NN1 Er_FU ._. 
What_DDQ happens_VVZ in_II the_AT &quot;_" steady_JJ current_JJ &quot;_" case_NN1 if_CS we_PPIS2 fill_VV0 the_AT whole_JJ space_NN1 by_II a_AT1 magnetic_JJ material_NN1 ?_? 
The_AT magnetic_JJ field_NN1 remains_VVZ unchanged_JJ and_CC the_AT magnetic_JJ flux_NN1 density_NN1 increases_VVZ by_II a_AT1 factor_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Does_VDZ the_AT analogy_NN1 still_RR hold_VVI when_RRQ the_AT magnetic_JJ material_NN1 fills_VVZ only_JJ part_NN1 of_IO the_AT space_NN1 ?_? 
The_AT answer_NN1 is_VBZ not_XX quite_RR ,_, because_CS the_AT boundary_NN1 conditions_NN2 are_VBR not_XX quite_RG the_AT same_DA ._. 
Using_VVG the_AT technique_NN1 (_( surface_NN1 and_CC line_NN1 integrals_NN2 shrinking_VVG to_TO zero_VVI )_) as_CSA in_II Section_NN1 2.10_MC we_PPIS2 get_VV0 the_AT boundary_NN1 conditions_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS K_ZZ1 is_VBZ a_AT1 surface_NN1 current_NN1 ._. 
Hence_RR in_II the_AT absence_NN1 of_IO surface_NN1 charges_NN2 and_CC surface_NN1 currents_NN2 the_AT analogy_NN1 still_RR holds_VVZ ._. 
If_CS we_PPIS2 look_VV0 at_II the_AT magnetic_JJ field_NN1 in_II a_AT1 region_NN1 in_II which_DDQ no_AT currents_NN2 flow_VV0 ,_, the_AT analogy_NN1 is_VBZ even_RR closer_RRR because_CS &lsqb;_( formula_NN1 &rsqb;_) and_CC we_PPIS2 can_VM introduce_VVI a_AT1 magnetic_JJ scalar_JJ potential_NN1 with_IW the_AT relation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
A_AT1 good_JJ example_NN1 is_VBZ the_AT calculation_NN1 of_IO the_AT effect_NN1 of_IO a_AT1 spherical_JJ piece_NN1 of_IO magnetic_JJ material_NN1 inserted_VVN into_II a_AT1 homogeneous_JJ magnetic_JJ field_NN1 ._. 
The_AT mathematics_NN1 is_VBZ exactly_RR the_AT same_DA as_CSA in_II the_AT electrostatic_JJ case_NN1 ;_; we_PPIS2 need_VV0 only_RR to_TO replace_VVI E0_FO by_II H0_FO and_CC Er_FU by_II &lsqb;_( formula_NN1 &rsqb;_) in_II eqn_NN1 (_( 2.110_MC )_) in_BCL21 order_BCL22 to_TO get_VVI &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR Fig._NN1 2.41_MC (_( p._NNU 47_MC )_) is_VBZ a_AT1 valid_JJ representation_NN1 for_IF magnetic_JJ materials_NN2 as_RR21 well_RR22 ._. 
The_AT difference_NN1 between_II the_AT two_MC cases_NN2 is_VBZ merely_RR quantitative_JJ ._. 
The_AT magnetic_JJ materials_NN2 used_VVN in_II practice_NN1 have_VH0 a_AT1 &lsqb;_( formula_NN1 &rsqb;_) several_DA2 orders_NN2 of_IO magnitude_NN1 higher_JJR than_CSN the_AT Er_NP1 of_IO practical_JJ dielectric_JJ materials_NN2 ._. 
Hence_RR the_AT &quot;_" attraction_NN1 &quot;_" of_IO the_AT field_NN1 lines_NN2 is_VBZ much_DA1 more_DAR pronounced_VVN in_II the_AT magnetic_JJ case_NN1 ._. 
To_TO give_VVI a_AT1 practical_JJ example_NN1 let_VV0 us_PPIO2 put_VVN in_II the_AT middle_NN1 of_IO our_APPGE solenoid_NN1 a_AT1 magnetic_JJ material_NN1 of_IO high_JJ &lsqb;_( formula_NN1 &rsqb;_) as_CSA shown_VVN in_II Fig._NN1 3.9_MC ._. 
Now_RT practically_RR all_DB the_AT flux_NN1 lines_NN2 are_VBR funnelled_VVN into_II the_AT magnetic_JJ material_NN1 so_RG much_DA1 so_CS21 that_CS22 we_PPIS2 are_VBR entitled_VVN to_TO regard_VVI both_RR H_ZZ1 and_CC B_ZZ1 as_CSA being_VBG zero_MC outside_II the_AT magnetic_JJ material._NNU 3.10_MC ._. 
Magnetic_JJ materials_NN2 When_CS studying_VVG the_AT electromagnetic_JJ properties_NN2 of_IO dielectrics_NN2 I_PPIS1 was_VBDZ very_RG reluctant_JJ to_TO get_VVI involved_JJ with_IW the_AT physics_NN1 of_IO dielectrics_NN2 ._. 
Now_RT I_PPIS1 am_VBM even_RR more_RGR reluctant_JJ to_TO talk_VVI about_II the_AT physics_NN1 of_IO magnetic_JJ materials_NN2 ._. 
Not_XX so_RG much_DA1 because_CS I_PPIS1 do_VD0 n't_XX understand_VVI the_AT subject_NN1 (_( that_DD1 is_VBZ no_AT real_JJ obstacle_NN1 to_II a_AT1 lecturer_NN1 )_) ,_, but_CCB more_RRR because_II21 of_II22 the_AT time_NNT1 we_PPIS2 are_VBR likely_JJ to_TO consume_VVI ,_, even_CS21 if_CS22 we_PPIS2 keep_VV0 a_AT1 respectable_JJ distance_NN1 from_II quantum_NN1 mechanics_NN2 and_CC concentrate_VV0 solely_RR on_II phenomenological_JJ theories_NN2 ._. 
It_PPH1 must_VM be_VBI admitted_VVN that_CST magnetic_JJ materials_NN2 behave_VV0 most_RGT unreasonably_RR ._. 
The_AT relative_JJ permeability_NN1 may_VM turn_VVI out_RP to_TO be_VBI a_AT1 tensor_NN1 ;_; it_PPH1 may_VM take_VVI on_RP very_RG high_JJ values_NN2 ,_, say_VV0 106_MC or_CC more_RRR ,_, or_CC it_PPH1 may_VM have_VHI a_AT1 value_NN1 very_RG close_JJ to_II unity_NN1 ,_, say_VV0 1.00002_MC for_IF a_AT1 wide_JJ range_NN1 of_IO temperatures_NN2 ,_, and_CC then_RT on_II cooling_VVG the_AT material_NN1 another_DD1 few_DA2 millidegrees_NN2 &lsqb;_( formula_NN1 &rsqb;_) may_VM drop_VVI to_TO zero_VVI ._. 
Why_RRQ to_TO bother_VVI so_RG much_DA1 about_II the_AT details_NN2 ,_, you_PPY may_VM ask_VVI ;_; could_VM n't_XX we_PPIS2 just_RR say_VVI that_CST &lsqb;_( formula_NN1 &rsqb;_) and_CC go_VVI on_II considering_VVG further_JJR examples_NN2 ?_? 
The_AT trouble_NN1 is_VBZ that_CST for_IF the_AT most_RGT widely_RR used_JJ magnetic_JJ material_NN1 iron_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ not_XX a_AT1 constant_JJ ._. 
Not_XX only_RR that_CST its_APPGE value_NN1 depends_VVZ on_II H_ZZ1 ,_, but_CCB ,_, even_RR worse_JJR ,_, it_PPH1 depends_VVZ on_II the_AT previous_JJ history_NN1 of_IO the_AT sample_NN1 ._. 
So_RR we_PPIS2 have_VH0 to_TO discuss_VVI the_AT B-H_NP1 curves_NN2 ,_, and_CC I_PPIS1 will_VM include_VVI a_AT1 few_DA2 more_DAR things_NN2 ,_, but_CCB you_PPY must_VM realize_VVI that_CST there_EX is_VBZ a_RR21 lot_RR22 more_RRR to_II it_PPH1 ._. 
By_II the_AT time_NNT1 we_PPIS2 finish_VV0 with_IW it_PPH1 we_PPIS2 shall_VM have_VHI just_RR scratched_VVN the_AT surface_NN1 of_IO the_AT subject._NNU 3.11_MC ._. 
The_AT B-H_NP1 curve_NN1 of_IO ferromagnetic_JJ materials_NN2 It_PPH1 is_VBZ customary_JJ to_TO divide_VVI all_DB magnetic_JJ materials_NN2 into_II diamagnetic_JJ (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ,_, paramagnetic_JJ (_( &lsqb;_( formula_NN1 &rsqb;_) )_) ,_, and_CC ferromagnetic_JJ (_( &lsqb;_( formula_NN1 &rsqb;_) )_) groups_NN2 ._. 
Paramagnetic_JJ materials_NN2 are_VBR unimportant_JJ from_II an_AT1 engineering_NN1 point_NN1 of_IO view_NN1 ,_, and_CC diamagnetic_JJ materials_NN2 may_VM have_VHI a_AT1 future_NN1 ;_; the_AT present_NN1 ,_, however_RR ,_, belongs_VVZ to_II the_AT ferromagnetic_JJ group_NN1 ,_, and_CC above_II all_DB to_II the_AT most_RGT important_JJ representative_NN1 of_IO the_AT group_NN1 the_AT various_JJ alloys_NN2 of_IO iron_NN1 ._. 
In_BCL21 order_BCL22 to_TO obtain_VVI the_AT B-H_NP1 curve_NN1 let_VV0 us_PPIO2 make_VVI the_AT following_JJ experiment_NN1 ._. 
Place_VV0 a_AT1 cylindrical_JJ iron_NN1 rod_NN1 inside_II a_AT1 solenoid_NN1 (_( as_CSA in_II Fig._NN1 3.9_MC )_) ,_, vary_VV0 the_AT current_JJ and_CC measure_VV0 the_AT flux_NN1 density_NN1 ._. 
We_PPIS2 shall_VM assume_VVI that_DD1 ,_, before_CS we_PPIS2 switch_VV0 on_RP the_AT current_JJ ,_, H_ZZ1 =_FO 0_MC and_CC B_ZZ1 =_FO 0_MC ,_, a_AT1 natural-enough_RR assumption_NN1 ._. 
As_CSA we_PPIS2 increase_VV0 the_AT current_JJ ,_, the_AT magnetic_JJ field_NN1 will_VM increase_VVI proportionally_RR (_( eqn_NN1 (_( 3.71_MC )_) )_) but_CCB the_AT flux_NN1 density_NN1 will_VM be_VBI a_AT1 nonlinear_JJ function_NN1 of_IO the_AT magnetic_JJ field_NN1 ,_, as_CSA shown_VVN by_II the_AT line_NN1 OP_NN1 in_II Fig._NN1 3.10_MC ._. 
The_AT point_NN1 P_ZZ1 is_VBZ called_VVN the_AT saturation_NN1 point_NN1 ,_, beyond_II which_DDQ &lsqb;_( formula_NN1 &rsqb;_) ,_, i.e._REX the_AT iron_NN1 has_VHZ stopped_VVN contributing_VVG to_II the_AT flux_NN1 density_NN1 ;_; any_DD further_JJR increase_NN1 of_IO H_ZZ1 will_VM result_VVI in_II that_DD1 much_DA1 increase_NN1 of_IO B_ZZ1 as_CSA in_II a_AT1 vacuum_NN1 ._. 
Reducing_VVG now_RT the_AT current_JJ (_( and_CC H_ZZ1 with_IW it_PPH1 )_) we_PPIS2 shall_VM not_XX retrace_VVI the_AT same_DA curve_NN1 ;_; B_ZZ1 will_VM decrease_VVI much_RR more_RGR slowly_RR and_CC will_VM have_VHI a_AT1 finite_JJ value_NN1 at_II zero_MC current_JJ ._. 
If_CS we_PPIS2 now_RT increase_VV0 the_AT current_JJ in_II the_AT opposite_JJ direction_NN1 B_ZZ1 will_VM decrease_VVI further_RRR reaching_VVG zero_NN1 at_II R_ZZ1 ,_, and_CC negative_JJ saturation_NN1 at_II S._NP1 The_AT other_JJ half_NN1 of_IO the_AT curve_NN1 STUP_NN1 displays_VVZ the_AT same_DA behaviour_NN1 ._. 
This_DD1 is_VBZ no_RR21 doubt_RR22 a_AT1 remarkable_JJ curve_NN1 ._. 
B_NP1 being_VBG so_RG sluggish_JJ ,_, it_PPH1 is_VBZ usually_RR referred_VVN to_II as_II the_AT hysteresis_NN1 curve_NN1 ._. 
Note_VV0 that_CST the_AT B-H_NP1 relationship_NN1 is_VBZ irreversible_JJ everywhere_RL inside_II the_AT hysteresis_NN1 curve_NN1 ._. 
If_CS at_II an_AT1 arbitrary_JJ point_NN1 (_( say_VV0 V_ZZ1 )_) we_PPIS2 decided_VVD to_TO decrease_VVI the_AT current_JJ ,_, B_ZZ1 would_VM decrease_VVI in_II a_AT1 different_JJ manner_NN1 ,_, as_CSA shown_VVN in_II Fig._NN1 3.10_MC ._. 
As_CS31 far_CS32 as_CS33 engineering_NN1 applications_NN2 are_VBR concerned_JJ the_AT most_RGT remarkable_JJ feature_NN1 of_IO the_AT curve_NN1 is_VBZ that_CST we_PPIS2 can_VM get_VVI a_AT1 flux_NN1 density_NN1 even_RR in_II the_AT absence_NN1 of_IO all_DB external_JJ agents_NN2 ._. 
This_DD1 is_VBZ of_RR21 course_RR22 the_AT permanent_JJ magnet_NN1 you_PPY have_VH0 all_DB come_VVI across_RL ._. 
How_RRQ is_VBZ this_DD1 possible_JJ ?_? 
Up_II21 to_II22 now_RT a_AT1 current_NN1 appeared_VVD to_TO be_VBI absolutely_RR essential_JJ for_IF creating_VVG a_AT1 magnetic_JJ flux_NN1 ._. 
The_AT obvious_JJ way_NN1 out_II21 of_II22 this_DD1 dilemma_NN1 is_VBZ to_TO say_VVI that_CST the_AT currents_NN2 are_VBR there_RL inside_II the_AT magnetic_JJ material_NN1 ._. 
The_AT detailed_JJ mechanism_NN1 of_IO these_DD2 currents_NN2 has_VHZ kept_VVN lots_PN of_IO physicists_NN2 busy_VV0 ever_RR since_CS Ampre_NP1 made_VVD the_AT hypothesis_NN1 ,_, but_CCB apparently_RR some_DD more_DAR time_NNT1 is_VBZ needed_VVN to_TO find_VVI a_AT1 proper_JJ solution_NN1 ._. 
For_IF permanent_JJ magnets_NN2 a_AT1 wide_JJ hysteresis_NN1 curve_NN1 is_VBZ needed_VVN so_CS21 that_CS22 demagnetization_NN1 should_VM not_XX easily_RR occur_VVI ._. 
For_IF transformers_NN2 and_CC rotating_VVG machinery_NN1 a_AT1 narrow_JJ hysteresis_NN1 curve_NN1 is_VBZ preferable_JJ because_CS irreversibility_NN1 leads_VVZ to_II losses_NN2 ._. 
I_PPIS1 am_VBM afraid_JJ this_DD1 is_VBZ about_RG as_RG much_DA1 as_CSA I_PPIS1 am_VBM going_VVGK to_TO say_VVI about_II magnetic_JJ materials_NN2 apart_II21 from_II22 a_AT1 very_RG brief_JJ discussion_NN1 of_IO some_DD pseudomagnetic_JJ materials_NN2 (_( superconductors_NN2 )_) in_II Sections_NN2 3.15_MC and_CC 3.16_MC ._. 
We_PPIS2 shall_VM look_VVI at_II a_AT1 few_DA2 examples_NN2 now_RT in_II which_DDQ simple_JJ geometries_NN2 will_VM be_VBI considered._NNU 3.12_MC The_AT magnetic_JJ flux_NN1 density_NN1 inside_II a_AT1 permanent_JJ magnet_NN1 of_IO toroidal_JJ shape_NN1 (_( B_ZZ1 )_) In_II a_AT1 permanent_JJ magnet_NN1 shaped_VVN as_II a_AT1 torus_NN1 we_PPIS2 have_VH0 H_ZZ1 =_FO 0_MC ._. 
The_AT only_JJ equation_NN1 to_TO satisfy_VVI is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC that_DD1 means_VVZ that_CST all_DB the_AT flux_NN1 lines_NN2 must_VM be_VBI closed_VVN ._. 
Choosing_VVG a_AT1 toroidal_JJ shape_NN1 there_EX are_VBR still_RR lots_PN of_IO possible_JJ ways_NN2 for_IF the_AT lines_NN2 to_TO arrange_VVI themselves_PPX2 ,_, but_CCB in_II a_AT1 good_JJ permanent_JJ magnet_NN1 the_AT flux_NN1 lines_NN2 will_VM be_VBI circles_NN2 with_IW centres_NN2 at_II 0_MC ,_, the_AT centre_NN1 of_IO the_AT toroid_NN1 (_( Fig._NN1 3.11_MC (_( a_ZZ1 )_) )_) ._. 
We_PPIS2 may_VM then_RT claim_VVI that_CST B_ZZ1 =_FO constant_JJ everywhere_RL inside_II the_AT magnet_NN1 and_CC zero_MC outside._NNU 3.13_MC ._. 
The_AT magnetic_JJ field_NN1 inside_II a_AT1 permanent_JJ magnet_NN1 of_IO toroidal_JJ shape_NN1 containing_VVG a_AT1 gap_NN1 &lsqb;_( H-B_NP1 )_) In_II the_AT last_MD section_NN1 we_PPIS2 have_VH0 come_VVN to_II the_AT interesting_JJ conclusion_NN1 that_CST B_ZZ1 may_VM alone_RR exist_VVI of_IO all_DB our_APPGE variables_NN2 but_CCB we_PPIS2 reached_VVD that_DD1 conclusion_NN1 on_II a_AT1 magnet_NN1 shape_NN1 not_XX much_RR used_VVN in_II practice_NN1 ._. 
If_CS we_PPIS2 take_VV0 the_AT trouble_NN1 to_TO make_VVI a_AT1 permanent_JJ magnet_NN1 we_PPIS2 would_VM like_VVI to_TO have_VHI access_NN1 to_II the_AT magnetic_JJ flux_NN1 so_RR let_VV0 us_PPIO2 look_VVI at_II the_AT more_RGR practical_JJ case_NN1 (_( Fig._NN1 3.11(b)_FO )_) when_RRQ a_AT1 narrow_JJ gap_NN1 is_VBZ cut_VVN into_II the_AT magnet_NN1 ._. 
How_RRQ will_VM the_AT flux_NN1 density_NN1 vary_VV0 in_II the_AT gap_NN1 ?_? 
It_PPH1 will_VM be_VBI hardly_RR different_JJ from_II B0_FO ,_, the_AT value_NN1 in_II the_AT material_NN1 ;_; in_II the_AT short_JJ space_NN1 available_JJ the_AT flux_NN1 lines_NN2 have_VH0 not_XX got_VVN a_AT1 chance_NN1 to_TO spread_VVI ._. 
We_PPIS2 shall_VM therefore_RR take_VVI the_AT flux_NN1 density_NN1 in_II the_AT gap_NN1 to_TO be_VBI equal_JJ to_II B0_FO ._. 
The_AT corresponding_JJ magnetic_JJ field_NN1 will_VM be_VBI &lsqb;_( formula_NN1 &rsqb;_) ._. 
What_DDQ about_II the_AT value_NN1 of_IO the_AT magnetic_JJ field_NN1 in_II the_AT magnet_NN1 ?_? 
It_PPH1 may_VM be_VBI obtained_VVN from_II the_AT consideration_NN1 that_CST in_II the_AT absence_NN1 of_IO an_AT1 external_JJ current_JJ the_AT line_NN1 integral_JJ of_IO H_ZZ1 (_( taken_VVN over_II the_AT dotted_JJ lines_NN2 in_II Fig._NN1 3.11(b)_FO )_) must_VM vanish_VVI ,_, leading_VVG to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT width_NN1 of_IO the_AT gap_NN1 and_CC l_ZZ1 is_VBZ the_AT length_NN1 of_IO the_AT path_NN1 in_II the_AT magnet_NN1 ._. 
From_II eqn_NN1 (_( 3.76_MC )_) we_PPIS2 get_VV0 the_AT value_NN1 of_IO the_AT magnetic_JJ field_NN1 inside_II the_AT magnet_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) ._. 
There_EX is_VBZ still_RR one_MC1 question_NN1 we_PPIS2 have_VH0 to_TO ask_VVI ,_, Will_VM the_AT magnetic_JJ flux_NN1 density_NN1 inside_II the_AT material_NN1 B0_FO be_VBI the_AT same_DA as_CSA Br_JJ the_AT value_NN1 before_II the_AT gap_NN1 was_VBDZ cut_VVN ?_? 
No_UH ,_, not_XX quite_RR ._. 
There_EX are_VBR now_RT two_MC relations_NN2 to_TO satisfy_VVI ._. 
B0_FO is_VBZ obtained_VVN where_CS the_AT straight_JJ line_NN1 &lsqb;_( formula_NN1 &rsqb;_) intersects_VVZ the_AT B-H_NP1 hysteresis_NN1 curve_NN1 as_CSA shown_VVN in_II Fig._NN1 3.12._MC 3.14_MC ._. 
The_AT magnetic_JJ field_NN1 in_II a_AT1 ferromagnetic_JJ material_NN1 of_IO toroidal_JJ shape_NN1 excited_VVN by_II a_AT1 steady_JJ current_JJ (_( J_ZZ1 ,_, H_ZZ1 ,_, B_ZZ1 )_) We_PPIS2 shall_VM now_RT take_VVI a_AT1 ferromagnetic_JJ material_NN1 that_CST has_VHZ a_AT1 very_RG narrow_JJ hysteresis_NN1 loop_NN1 so_CS21 that_CS22 we_PPIS2 can_VM assume_VVI with_IW good_JJ approximation_NN1 a_AT1 unique_JJ (_( though_CS of_RR21 course_RR22 nonlinear_JJ )_) relationship_NN1 between_II B_ZZ1 and_CC H._NP1 The_AT material_NN1 is_VBZ again_RT assumed_VVN to_TO be_VBI of_IO a_AT1 toroidal_JJ shape_NN1 but_CCB it_PPH1 is_VBZ now_RT excited_VVN by_II a_AT1 current_JJ I_PPIS1 flowing_VVG in_II a_AT1 coil_NN1 of_IO N_ZZ1 turns_VVZ (_( Fig._NN1 3.13(a)_FO )_) ._. 
We_PPIS2 may_VM then_RT apply_VVI Ampre_NP1 's_GE law_NN1 to_II the_AT path_NN1 shown_VVN by_II dotted_JJ lines_NN2 to_TO obtain_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, whence_RRQ &lsqb;_( formula_NN1 &rsqb;_) ,_, a_AT1 formula_NN1 we_PPIS2 have_VH0 already_RR met_VVN (_( eqn_NN1 (_( 3.71_MC )_) )_) when_CS discussing_VVG an_AT1 approximate_JJ solution_NN1 for_IF a_AT1 long_JJ solenoid_NN1 ._. 
The_AT corresponding_JJ value_NN1 of_IO the_AT magnetic_JJ flux_NN1 density_NN1 is_VBZ B1_FO =_FO B(H1)_FO that_DD1 can_VM be_VBI obtained_VVN from_II the_AT B-H_NP1 curve_NN1 ,_, as_CSA shown_VVN in_II Fig._NN1 3.14_MC ._. 
Next_MD ,_, we_PPIS2 shall_VM find_VVI the_AT solution_NN1 when_CS there_EX is_VBZ a_AT1 gap_NN1 of_IO width_NN1 &lsqb;_( formula_NN1 &rsqb;_) in_II the_AT material_NN1 (_( Fig._NN1 3.13(b)_FO )_) ._. 
For_IF the_AT same_DA current_JJ the_AT magnetic_JJ flux_NN1 density_NN1 will_VM decrease_VVI to_II B2_FO ,_, its_APPGE value_NN1 being_VBG the_AT same_DA both_RR in_II air_NN1 and_CC in_II the_AT material_NN1 ._. 
Denoting_VVG the_AT magnetic_JJ field_NN1 in_II the_AT material_NN1 by_II H2i_FO and_CC in_II air_NN1 by_II H20_FO ,_, Ampre_NP1 's_GE law_NN1 yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Noting_VVG further_RRR that_DD1 &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT above_JJ equation_NN1 gives_VVZ a_AT1 linear_JJ relationship_NN1 between_II H2i_FO and_CC B2_FO ._. 
Hence_RR the_AT solution_NN1 is_VBZ obtained_VVN by_II the_AT intersection_NN1 of_IO the_AT straight_JJ line_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT B-H_NP1 curve_NN1 ._. 
The_AT graphical_JJ construction_NN1 is_VBZ shown_VVN in_II Fig._NN1 3.14_MC ._. 
If_CS we_PPIS2 wish_VV0 to_TO produce_VVI the_AT same_DA flux_NN1 density_NN1 as_CSA in_II the_AT absence_NN1 of_IO the_AT gap_NN1 ,_, we_PPIS2 need_VV0 to_TO increase_VVI the_AT current_JJ ._. 
This_DD1 is_VBZ the_AT subject_NN1 of_IO Example_NN1 3.13._MC 3.15_MC The_AT perfect_JJ diamagnet_NN1 (_( H_ZZ1 ,_, B_ZZ1 ,_, K_ZZ1 )_) There_EX is_VBZ a_AT1 class_NN1 of_IO materials_NN2 called_VVN type_NN1 I_ZZ1 superconductors_NN2 which_DDQ below_II a_AT1 certain_JJ critical_JJ temperature_NN1 (_( around_II the_AT normal_JJ boiling-point_NN1 of_IO liquid_JJ helium_NN1 &lsqb;_( formula_NN1 &rsqb;_) )_) become_VV0 both_RR perfect_JJ conductors_NN2 and_CC perfect_JJ diamagnets_NN2 ._. 
This_DD1 means_VVZ that_CST B_ZZ1 must_VM be_VBI zero_NN1 inside_II the_AT material_NN1 ._. 
Thus_RR if_CS we_PPIS2 place_VV0 a_AT1 piece_NN1 of_IO such_DA material_NN1 (_( say_VV0 a_AT1 sphere_NN1 )_) into_II an_AT1 otherwise_RR constant_JJ magnetic_JJ flux_NN1 ,_, the_AT material_NN1 will_VM expel_VVI the_AT flux_NN1 lines_NN2 as_CSA shown_VVN in_II Fig._NN1 3.15_MC ._. 
How_RRQ can_VM this_DD1 happen_VVI ?_? 
What_DDQ mechanism_NN1 is_VBZ responsible_JJ for_IF expelling_VVG the_AT magnetic_JJ flux_NN1 ?_? 
The_AT appearance_NN1 of_IO surface_NN1 currents_NN2 (_( which_DDQ we_PPIS2 have_VH0 previously_RR denoted_VVN by_II K_ZZ1 )_) ._. 
As_II the_AT sphere_NN1 becomes_VVZ superconducting_JJ ,_, surface_NN1 currents_NN2 are_VBR set_VVN up_RP producing_VVG a_AT1 magnetic_JJ flux_NN1 opposite_RL to_II that_DD1 already_RR existing_JJ inside_II the_AT material._NNU 3.16_MC ._. 
The_AT penetration_NN1 of_IO the_AT magnetic_JJ flux_NN1 into_II a_AT1 type_NN1 I_MC1 superconductor_NN1 (_( J_ZZ1 ,_, H_ZZ1 ,_, B_ZZ1 )_) In_II real_JJ life_NN1 nothing_PN1 is_VBZ ever_RR perfect_JJ ._. 
A_AT1 type_NN1 I_MC1 superconductor_NN1 ,_, I_PPIS1 confess_VV0 ,_, is_VBZ not_XX a_AT1 perfect_JJ diamagnet_NN1 ;_; it_PPH1 will_VM let_VVI in_II the_AT magnetic_JJ flux_NN1 just_RR a_AT1 little_JJ bit_NN1 ._. 
How_RGQ far_RR will_VM the_AT magnetic_JJ flux_NN1 penetrate_VV0 ?_? 
Is_VBZ there_EX a_AT1 simple_JJ way_NN1 of_IO describing_VVG the_AT decay_NN1 of_IO the_AT magnetic_JJ flux_NN1 mathematically_RR ?_? 
There_EX is_VBZ ._. 
We_PPIS2 can_VM use_VVI the_AT following_JJ one-dimensional_JJ model_NN1 ._. 
Half_DB of_IO the_AT space_NN1 (_( z&lt;0_FO )_) is_VBZ a_AT1 vacuum_NN1 in_II which_DDQ a_AT1 constant_JJ flux_NN1 density_NN1 B_ZZ1 =_FO B0iy_FO is_VBZ assumed_VVN ._. 
The_AT corresponding_JJ vector_NN1 potential_NN1 (_( assuming_VVG no_AT variation_NN1 in_II the_AT transverse_JJ direction_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS A0_FO and_CC A1_FO are_VBR constants_NN2 ._. 
The_AT other_JJ half_NN1 of_IO the_AT space_NN1 (_( z&gt;0_FO )_) is_VBZ filled_VVN with_IW a_AT1 type_NN1 I_MC1 superconductor_NN1 ._. 
The_AT approach_NN1 I_PPIS1 am_VBM going_VVGK to_TO adopt_VVI now_RT applies_VVZ to_II the_AT present_JJ section_NN1 only_RR and_CC acknowledges_VVZ the_AT fact_NN1 that_CST a_AT1 superconductor_NN1 is_VBZ not_XX an_AT1 &quot;_" ordinary_JJ &quot;_" magnetic_JJ material_NN1 ;_; it_PPH1 can_VM not_XX be_VBI described_VVN by_II a_AT1 &quot;_" magnetic_JJ &quot;_" constant_JJ ,_, by_II assigning_VVG to_II it_PPH1 a_AT1 certain_JJ value_NN1 for_IF &lsqb;_( formula_NN1 &rsqb;_) ._. 
Instead_RR ,_, a_AT1 new_JJ macroscopic_JJ constant_JJ needs_NN2 to_TO be_VBI introduced_VVN ._. 
We_PPIS2 shall_VM proceed_VVI similarly_RR as_CSA in_II the_AT case_NN1 of_IO conductors_NN2 ._. 
We_PPIS2 assumed_VVD then_RT that_CST the_AT electric_JJ field_NN1 was_VBDZ proportional_JJ to_II the_AT current_JJ density_NN1 ._. 
Now_RT we_PPIS2 shall_VM assume_VVI that_CST there_EX is_VBZ a_AT1 linear_JJ relationship_NN1 between_II the_AT vector_NN1 potential_NN1 and_CC the_AT current_JJ density_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS y_ZZ1 is_VBZ the_AT new_JJ macroscopic_JJ constant_JJ ._. 
Substituting_VVG eqn_NN1 (_( 3.84_MC )_) into_II (_( 3.11_MC )_) and_CC retaining_VVG the_AT one-dimensional_JJ character_NN1 of_IO the_AT problem_NN1 ,_, we_PPIS2 get_VV0 the_AT differential_JJ equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT relevant_JJ solution_NN1 of_IO which_DDQ (_( disregarding_VVG the_AT exponentially_RR increasing_JJ term_NN1 )_) is_VBZ as_CSA follows_VVZ :_: &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS A2_FO is_VBZ another_DD1 constant_JJ ._. 
Hence_RR we_PPIS2 get_VV0 for_IF the_AT magnetic_JJ flux_NN1 density_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT flux_NN1 density_NN1 reduces_VVZ to_II 1/e_FU of_IO its_APPGE value_NN1 at_II a_AT1 distance_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ is_VBZ called_VVN the_AT penetration_NN1 depth_NN1 ._. 
A_AT1 typical_JJ value_NN1 is_VBZ =_FO 60_MC nm_FU (_( nm_FU =_FO nanometre_NNU1 )_) ,_, so_CS the_AT material_NN1 is_VBZ not_XX very_RG far_RR from_II being_VBG a_AT1 perfect_JJ diamagnet._NNU 3.17_MC ._. 
Forces_NN2 Let_VV0 us_PPIO2 first_MD work_VVI out_RP the_AT force_NN1 upon_II a_AT1 current_JJ element_NN1 of_IO length_NN1 ds_MC2 in_II the_AT presence_NN1 of_IO a_AT1 flux_NN1 density_NN1 B._NP1 According_II21 to_II22 eqn_NN1 (_( 3.7_MC )_) the_AT force_NN1 on_II a_AT1 point_NN1 charge_NN1 q_ZZ1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Assuming_VVG now_CS21 that_CS22 the_AT magnetic_JJ field_NN1 is_VBZ constant_JJ over_II S0_FO ,_, the_AT cross-section_NN1 of_IO the_AT wire_NN1 ,_, the_AT force_NN1 upon_II all_DB charges_NN2 within_II the_AT volume_NN1 element_NN1 S0ds_FO must_VM be_VBI the_AT same_DA ._. 
Hence_RR the_AT total_JJ force_NN1 on_II the_AT current_JJ element_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC for_IF a_AT1 whole_JJ current_JJ loop_NN1 we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS the_AT magnetic_JJ flux_NN1 density_NN1 is_VBZ constant_JJ over_II the_AT whole_JJ loop_NN1 then_RT B_ZZ1 may_VM be_VBI taken_VVN out_II21 of_II22 the_AT integral_JJ sign_NN1 ,_, and_CC the_AT remaining_JJ integration_NN1 yields_VVZ zero_MC ._. 
Consequently_RR ,_, there_EX is_VBZ no_AT net_JJ force_NN1 upon_II a_AT1 current_JJ loop_NN1 in_II a_AT1 homogeneous_JJ magnetic_JJ flux_NN1 ._. 
There_EX is_VBZ ,_, however_RR ,_, a_AT1 torque_NN1 which_DDQ may_VM be_VBI easily_RR calculated_VVN for_IF a_AT1 rectangular_JJ loop_NN1 ._. 
In_II the_AT specific_JJ example_NN1 of_IO Fig._NN1 3.16_MC (_( your_APPGE guess_NN1 is_VBZ correct_JJ if_CS you_PPY think_VV0 that_CST the_AT arrangement_NN1 has_VHZ something_PN1 to_TO do_VDI with_IW electrical_JJ machines_NN2 )_) there_EX is_VBZ a_AT1 current-carrying_JJ loop_NN1 capable_JJ to_TO rotate_VVI around_II the_AT horizontal_JJ axis_NN1 in_II the_AT magnetic_JJ flux_NN1 (_( assumed_VVN constant_JJ )_) of_IO the_AT permanent_JJ magnet_NN1 ._. 
As_CSA may_VM be_VBI seen_VVN from_II Fig._NN1 3.16(b)_FO the_AT forces_NN2 on_II sides_NN2 1_MC1 and_CC 3_MC balance_VV0 each_PPX221 other_PPX222 ,_, whereas_CS those_DD2 acting_VVG upon_II sides_NN2 2_MC and_CC 4_MC produce_VV0 a_AT1 torque_NN1 ._. 
The_AT force_NN1 on_II side_NN1 2_MC is_VBZ &lsqb;_( formula_NN1 &rsqb;_) whence_RRQ the_AT torque_NN1 comes_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) where_RRQ 0_MC is_VBZ the_AT angle_NN1 between_II the_AT plane_NN1 of_IO the_AT loop_NN1 and_CC the_AT vertical_JJ direction_NN1 ._. 
Next_MD we_PPIS2 shall_VM work_VVI out_RP the_AT force_NN1 upon_II a_AT1 small_JJ current_JJ loop_NN1 due_II21 to_II22 another_DD1 current_JJ loop_NN1 in_II the_AT geometry_NN1 of_IO Fig._NN1 3.17_MC ._. 
Owing_II21 to_II22 symmetry_NN1 the_AT magnetic_JJ field_NN1 of_IO loop_NN1 I_ZZ1 has_VHZ only_RR Hz_NNU and_CC HR_NNU components_NN2 ._. 
In_II the_AT vicinity_NN1 of_IO the_AT z_ZZ1 axis_NN1 they_PPHS2 are_VBR given_VVN by_II eqn_NN1 (_( 3.65_MC )_) ._. 
We_PPIS2 shall_VM consider_VVI the_AT forces_NN2 due_II21 to_II22 each_DD1 component_NN1 separately_RR ._. 
The_AT force_NN1 due_II21 to_II22 Hz_NNU upon_II the_AT current_JJ element_NN1 in_II loop_NN1 2_MC is_VBZ in_II the_AT radial_JJ direction_NN1 ._. 
Integrating_VVG over_RP all_DB current_JJ elements_NN2 the_AT net_JJ force_NN1 is_VBZ obviously_RR zero_MC ._. 
For_IF the_AT other_JJ component_NN1 ,_, HR_NNU the_AT force_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ in_II the_AT negative_JJ z_ZZ1 direction_NN1 ._. 
(_( Remember_VV0 from_II school_NN1 ?_? 
&quot;_" The_AT force_NN1 between_II two_MC currents_NN2 flowing_VVG in_II the_AT same_DA direction_NN1 is_VBZ attractive_JJ ._. 
&quot;_" )_) Integration_NN1 over_II the_AT circumference_NN1 yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Examples_NN2 3_MC 3.1_MC ._. 
A_AT1 current_JJ with_IW a_AT1 constant_JJ current_JJ density_NN1 J0_FO flows_VVZ in_II the_AT z-direction_NN1 between_II the_AT cylinders_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Determine_VV0 the_AT magnetic_JJ field_NN1 in_II the_AT regions_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
3.2_MC ._. 
A_AT1 cylindrical_JJ column_NN1 of_IO mercury_NN1 10mm_NNU diameter_NN1 carries_VVZ a_AT1 current_JJ of_IO 100_MC A_ZZ1 uniformly_RR distributed_VVN over_II the_AT cross-section_NN1 ._. 
Calculate_VV0 the_AT pressure_NN1 due_II21 to_II22 the_AT pinch_NN1 effect_NN1 ,_, (_( i_ZZ1 )_) at_II a_AT1 radius_NN1 of_IO 2.5_MC mm_NNU and_CC (_( ii_MC )_) at_II the_AT axis_NN1 of_IO the_AT conductor._NNU 3.3_MC ._. 
The_AT resistance_NN1 between_II a_AT1 pair_NN of_IO electrodes_NN2 immersed_VVN in_II an_AT1 infinite_JJ medium_NN1 of_IO conductivity_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ R._NP1 Show_VV0 that_CST the_AT capacitance_NN1 between_II the_AT same_DA pair_NN of_IO electrodes_NN2 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) when_RRQ the_AT medium_NN1 is_VBZ changed_VVN to_II a_AT1 lossless_JJ dielectric_JJ ._. 
A_AT1 general_JJ proof_NN1 is_VBZ required_VVN valid_JJ for_IF any_DD geometry._NNU 3.4_MC ._. 
Two_MC lossy_JJ dielectric_JJ materials_NN2 are_VBR joined_VVN together_RL as_CSA shown_VVN in_II Fig._NN1 3.18._MC determine_VV0 the_AT voltages_NN2 across_II each_DD1 material_NN1 and_CC the_AT surface_NN1 charge_NN1 density_NN1 at_II the_AT boundary_NN1 if_CS a_AT1 voltage_NN1 V_ZZ1 is_VBZ applied._NNU 3.5_MC ._. 
Solve_VV0 eqn_NN1 (_( 3.47_MC )_) under_II the_AT condition_NN1 that_CST the_AT electric_JJ field_NN1 is_VBZ zero_MC at_II electrode_NN1 1_MC1 (_( z_ZZ1 =_FO 0_MC )_) ._. 
3.6_MC ._. 
Derive_VV0 an_AT1 expression_NN1 for_IF the_AT magnetic_JJ field_NN1 H_ZZ1 at_II a_AT1 point_NN1 P_ZZ1 distant_JJ a_AT1 from_II the_AT centre_NN1 line_NN1 of_IO a_AT1 long_JJ thin_JJ conducting_NN1 strip_NN1 of_IO width_NN1 b_ZZ1 (_( Fig._NN1 3.19_MC )_) which_DDQ carries_VVZ a_AT1 longitudinal_JJ current_JJ I_PPIS1 uniformly_RR distributed_VVN across_II its_APPGE section._NNU 3.7._MC (_( i_ZZ1 )_) Prove_VV0 that_CST the_AT magnetic_JJ field_NN1 strength_NN1 H_ZZ1 at_II a_AT1 point_NN1 P_ZZ1 distance_NN1 R_ZZ1 from_II the_AT wire_NN1 of_IO finite_JJ length_NN1 a_AT1 (_( Fig._NN1 3.20_MC )_) is_VBZ (_( ii_MC )_) Deduce_VV0 that_CST the_AT field_NN1 strength_NN1 at_II the_AT centre_NN1 of_IO a_AT1 square_JJ coil_NN1 of_IO side_NN1 b_ZZ1 ,_, carrying_VVG a_AT1 current_JJ I_ZZ1 ,_, is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
(_( iii_MC )_) Determine_VV0 the_AT field_NN1 strength_NN1 at_II the_AT centre_NN1 of_IO a_AT1 current_JJ carrying_NN1 loop_NN1 which_DDQ has_VHZ the_AT form_NN1 of_IO an_AT1 n-sided_JJ regular_JJ polygon_NN1 inscribed_VVN in_II a_AT1 circle_NN1 of_IO radius_NN1 a._NNU (_( iv_MC )_) Show_VV0 that_CST the_AT above_JJ result_NN1 reduces_VVZ to_II that_DD1 of_IO (_( ii_MC )_) when_RRQ n_ZZ1 =_FO 4_MC and_CC to_II eqn_NN1 (_( 3.65_MC )_) (_( with_IW z_ZZ1 =_FO 0_MC )_) when_RRQ n_ZZ1 </w>_NULL ._. 
3.8_MC ._. 
With_IW the_AT aid_NN1 of_IO the_AT transformation_NN1 &lsqb;_( formula_NN1 &rsqb;_) bring_VV0 eqn_NN1 (_( 3.55_MC )_) to_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) ._. 
3.9_MC ._. 
The_AT complete_JJ elliptic_JJ integrals_NN2 are_VBR defined_VVN as_CSA follows_VVZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Express_VV0 the_AT vector_NN1 potential_NN1 of_IO the_AT previous_JJ example_NN1 in_II31 terms_II32 of_II33 elliptic_JJ integrals_NN2 and_CC work_VV0 out_RP its_APPGE value_NN1 when_CS R_ZZ1 =_FO a_ZZ1 =_FO 0.1_MC m_NNO ,_, I_ZZ1 =_FO 1A_FO ,_, z_ZZ1 =_FO a_AT1 /12_MF (_( use_VV0 Tables_NN2 ,_, e.g._REX Jahnke-Emde-Losch_NP1 )_) ._. 
3.10_MC ._. 
When_CS k1_FO the_AT complete_JJ elliptic_JJ integrals_NN2 may_VM be_VBI approximated_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC &lsqb;_( formula_NN1 &rsqb;_) ._. 
Show_VV0 that_CST the_AT vector_NN1 potential_NN1 obtained_VVN in_II Example_NN1 3.9_MC reduces_VVZ to_II those_DD2 of_IO eqns_NN2 (_( 3.60_MC )_) and_CC (_( 3.64_MC )_) under_II the_AT respective_JJ assumptions._NNU 3.11_MC ._. 
It_PPH1 is_VBZ fairly_RR simple_JJ to_TO differentiate_VVI the_AT elliptic_JJ integrals_NN2 (_( see_VV0 p._NNU 49_MC of_IO Jahnke-Emde-Losch_NP1 ,_, for_IF the_AT formulae_NN2 )_) so_RR if_CS you_PPY have_VH0 the_AT patience_NN1 derive_VV0 the_AT magnetic_JJ field_NN1 from_II the_AT vector_NN1 potential_NN1 of_IO Example_NN1 3.9_MC and_CC determine_VV0 its_APPGE value_NN1 for_IF the_AT data_NN given_VVN there._NNU 3.12_MC ._. 
Eqn_NN1 (_( 3.70_MC )_) gives_VVZ the_AT magnetic_JJ field_NN1 on_II the_AT axis_NN1 of_IO a_AT1 solenoid_NN1 of_IO length_NN1 I._NP1 Is_VBZ the_AT formula_NN1 valid_JJ outside_II the_AT solenoid_NN1 ?_? 3.13_MC ._. 
The_AT magnetization_NN1 curve_NN1 of_IO a_AT1 certain_JJ steel_NN1 is_VBZ given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
An_AT1 anchor_NN1 ring_NN1 of_IO this_DD1 material_NN1 has_VHZ a_AT1 mean_JJ diameter_NN1 of_IO 255_MC mm_NNU ._. 
It_PPH1 is_VBZ wound_VVN with_IW 160_MC turns_NN2 of_IO wire_NN1 carrying_VVG a_AT1 current_JJ of_IO 2.5_MC A._NNU Calculate_VV0 the_AT flux_NN1 density_NN1 on_II the_AT mean_JJ diameter_NN1 ._. 
What_DDQ current_NN1 is_VBZ required_VVN to_TO produce_VVI the_AT same_DA flux_NN1 density_NN1 after_II a_AT1 gap_NN1 of_IO 0.5_MC mm_NNU wide_JJ has_VHZ been_VBN cut_VVN in_II the_AT ring_NN1 ?_? 
What_DDQ flux_NN1 density_NN1 is_VBZ produced_VVN in_II the_AT gap_NN1 by_II a_AT1 current_JJ of_IO 7.5_MC A_ZZ1 ?_? 3.14_MC ._. 
The_AT electromagnet_NN1 shown_VVN in_II section_NN1 in_II Fig._NN1 3.21_MC is_VBZ designed_VVN to_TO give_VVI radial_JJ magnetic_JJ flux_NN1 density_NN1 B_ZZ1 in_II an_AT1 annulus_NN1 of_IO radius_NN1 a_AT1 and_CC width_NN1 &lsqb;_( formula_NN1 &rsqb;_) when_CS energized_VVN with_IW constant_JJ voltage_NN1 V._II Its_APPGE coil_NN1 is_VBZ wound_VVN from_II copper_NN1 of_IO conductivity_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC is_VBZ located_VVN in_II the_AT annular_JJ space_NN1 inside_II the_AT electromagnet_NN1 which_DDQ has_VHZ the_AT dimensions_NN2 a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 shown_VVN in_II the_AT figure_NN1 ._. 
The_AT copper_NN1 can_VM be_VBI assumed_VVN to_TO be_VBI uniformly_RR distributed_VVN across_II the_AT section_NN1 but_CCB it_PPH1 only_RR occupies_VVZ a_AT1 fraction_NN1 &lsqb;_( formula_NN1 &rsqb;_) of_IO the_AT space_NN1 available_JJ ;_; the_AT current_JJ density_NN1 in_II the_AT copper_NN1 is_VBZ to_TO be_VBI J._NP1 Neglecting_VVG flux_NN1 leakage_NN1 ,_, and_CC the_AT contribution_NN1 of_IO the_AT magnetic_JJ material_NN1 to_II the_AT line_NN1 integral_JJ ,_, calculate_VV0 the_AT dimension_NN1 b_ZZ1 ,_, the_AT size_NN1 of_IO the_AT copper_NN1 wire_NN1 ,_, and_CC the_AT number_NN1 of_IO turns_NN2 in_II the_AT winding_JJ when_CS B_ZZ1 =_FO 1_MC1 T_ZZ1 ,_, a_ZZ1 =_FO 20_MC mm_NNU ,_, &lsqb;_( formula_NN1 &rsqb;_) =_FO 1.5_MC mm_NNU ,_, c_ZZ1 =_FO 0.1_MC m_NNO ,_, V_ZZ1 =_FO 12_MC V_NNU ,_, &lsqb;_( formula_NN1 &rsqb;_) =_FO 5.7x107_FO Sm-1_MC1 ,_, &lsqb;_( formula_NN1 &rsqb;_) =_FO 0.65_MC ,_, J_ZZ1 =1.5x106_FO Am-2_MC ._. 
Explain_VV0 why_RRQ &lsqb;_( formula_NN1 &rsqb;_) must_VM be_VBI less_DAR than_CSN unity_NN1 and_CC why_RRQ the_AT current_JJ density_NN1 must_VM be_VBI limited_VVN to_II J._NP1 Estimate_NN1 values_NN2 for_IF the_AT dimensions_NN2 h1_FO ,_, h2_FO so_CS21 that_CS22 each_DD1 section_NN1 of_IO the_AT steel_NN1 core_NN1 shall_VM be_VBI subject_II21 to_II22 approximately_RR the_AT same_DA maximum_JJ flux_NN1 density._NNU 3.15_MC ._. 
From_II eqn_NN1 (_( 3.90_MC )_) and_CC from_II the_AT Biot-Savart_NP1 law_NN1 show_VV0 that_CST the_AT force_NN1 between_II two_MC arbitrary_JJ current-carrying_JJ loops_NN2 (_( Fig._NN1 4.8_MC ,_, p._NN1 100_MC )_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ a_AT1 unit_NN1 vector_NN1 in_II the_AT direction_NN1 between_II the_AT current_JJ elements_NN2 ds1_FO and_CC ds2_FO ._. 
(_( Hint_NN1 ;_; Use_VV0 Stokes_NP1 's_GE theorem._NNU )_) 4_MC ._. 
Slowly_RR varying_JJ phenomena_NN2 4.1_MC ._. 
The_AT basic_JJ equations_NN2 IT_PPH1 is_VBZ difficult_JJ to_TO say_VVI what_DDQ &quot;_" slowly_RR &quot;_" varying_VVG is_VBZ until_CS I_PPIS1 give_VV0 examples_NN2 of_IO &quot;_" fast_JJ &quot;_" varying_JJ phenomena_NN2 ._. 
Thus_RR for_IF a_AT1 proper_JJ appreciation_NN1 of_IO the_AT distinction_NN1 between_II &quot;_" slow_JJ &quot;_" and_CC &quot;_" fast_RR &quot;_" you_PPY have_VH0 to_TO wait_VVI for_IF electromagnetic_JJ waves_NN2 to_TO be_VBI introduced_VVN ,_, discussed_VVD ,_, and_CC digested_VVN ._. 
For_IF immediate_JJ use_NN1 I_PPIS1 offer_VV0 only_JJ mathematics_NN1 ,_, but_CCB a_RR21 little_RR22 later_RRR (_( Section_NN1 4.3_MC )_) I_PPIS1 shall_VM try_VVI to_TO show_VVI the_AT limitations_NN2 imposed_VVN by_II the_AT assumption_NN1 of_IO slow_JJ variation_NN1 ._. 
Mathematically_RR ,_, the_AT definition_NN1 is_VBZ easy_JJ ._. 
As_CS31 long_CS32 as_CS33 &lsqb;_( formula_NN1 &rsqb;_) ,_, i.e._REX as_CS31 long_CS32 as_CS33 the_AT displacement_NN1 current_NN1 is_VBZ negligible_JJ in_II31 comparison_II32 with_II33 the_AT current_NN1 of_IO charged_JJ particles_NN2 ,_, we_PPIS2 are_VBR in_II the_AT &quot;_" slowly_RR &quot;_" varying_JJ region_NN1 ._. 
Hence_RR we_PPIS2 disregard_VV0 the_AT displacement_NN1 current_JJ term_NN1 but_CCB have_VH0 all_DB the_AT rest_NN1 of_IO eqns_NN2 (_( 1.1_MC )_) to_II 1.7_MC )_) as_CSA follows_VVZ :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
There_EX is_VBZ only_RR one_MC1 equation_NN1 we_PPIS2 have_VH0 not_XX considered_VVN so_RG far_RR and_CC that_DD1 is_VBZ eqn_NN1 (_( 4.3_MC )_) ,_, which_DDQ will_VM probably_RR look_VVI more_RGR familiar_JJ in_II another_DD1 form_NN1 ._. 
Integrating_VVG eqn_NN1 (_( 4.3_MC )_) over_II a_AT1 surface_NN1 and_CC applying_VVG Stokes_NP1 's_GE theorem_NN1 to_II the_AT left-hand_JJ side_NN1 ,_, we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS the_AT line_NN1 integral_JJ is_VBZ over_II the_AT closed_JJ contour_NN1 of_IO the_AT chosen_JJ surface_NN1 ._. 
Using_VVG the_AT definition_NN1 of_IO magnetic_JJ flux_NN1 (_( eqn_NN1 (_( 3.22_MC )_) )_) the_AT above_JJ equation_NN1 may_VM be_VBI written_VVN in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Remember_VV0 eqn_NN1 (_( 2.5_MC )_) &lsqb;_( formula_NN1 &rsqb;_) ,_, the_AT definition_NN1 of_IO the_AT scalar_JJ potential_NN1 in_II31 terms_II32 of_II33 the_AT electric_JJ field_NN1 ._. 
Substituting_VVG it_PPH1 into_II eqn_NN1 (_( 4.10_MC )_) would_VM always_RR yield_VVI zero_NN1 ,_, and_CC that_DD1 is_VBZ obviously_RR incorrect_JJ because_CS the_AT right-hand_JJ side_NN1 may_VM be_VBI finite_JJ ._. 
Thus_RR the_AT definition_NN1 of_IO eqn_NN1 (_( 2.5_MC )_) is_VBZ no_RR21 longer_RR22 applicable_JJ or_CC we_PPIS2 should_VM rather_RR say_VVI it_PPH1 is_VBZ no_RR21 longer_RR22 sufficient_JJ ._. 
We_PPIS2 need_VV0 something_PN1 else_RR besides_II the_AT scalar_JJ potential_NN1 for_IF describing_VVG correctly_RR a_AT1 time-varying_JJ electric_JJ field_NN1 ._. 
The_AT additional_JJ term_NN1 may_VM be_VBI obtained_VVN by_II substituting_VVG the_AT vector_NN1 potential_NN1 for_IF B_ZZ1 in_II eqn_NN1 (_( 4.3_MC )_) :_: &lsqb;_( formula_NN1 &rsqb;_) and_CC rearranging_VVG it_PPH1 in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF the_AT above_JJ equation_NN1 to_TO be_VBI satisfied_VVN the_AT expression_NN1 in_II the_AT bracket_NN1 must_VM be_VBI equal_JJ to_TO zero_VVI apart_II21 from_II22 the_AT gradient_NN1 of_IO a_AT1 scalar_JJ function_NN1 (_( remember_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Thus_RR the_AT general_JJ form_NN1 for_IF the_AT electric_JJ field_NN1 is_VBZ as_CSA follows_VVZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT gradient_NN1 of_IO a_AT1 scalar_JJ potential_NN1 is_VBZ still_RR there_RL but_CCB we_PPIS2 have_VH0 in_RR21 addition_RR22 the_AT time_NNT1 derivative_NN1 of_IO the_AT vector_NN1 potential_NN1 ._. 
When_CS studying_VVG static_JJ electricity_NN1 there_EX was_VBDZ no_AT need_NN1 to_TO make_VVI any_DD distinction_NN1 between_II voltage_NN1 and_CC potential_JJ difference_NN1 ._. 
For_IF the_AT static_JJ case_NN1 ,_, by_II definition_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF the_AT time-varying_JJ case_NN1 we_PPIS2 still_RR define_VV0 voltage_NN1 by_II the_AT relation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, but_CCB now_RT it_PPH1 takes_VVZ the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
There_EX is_VBZ no_AT reason_NN1 why_RRQ the_AT line_NN1 integral_JJ of_IO the_AT vector_NN1 potential_NN1 between_II two_MC arbitrary_JJ points_NN2 should_VM be_VBI independent_JJ of_IO the_AT path_NN1 ._. 
This_DD1 must_VM be_VBI kept_VVN in_II mind_NN1 when_CS considering_VVG time-varying_JJ fields_NN2 ._. 
It_PPH1 is_VBZ no_RR21 longer_RR22 unambiguous_JJ to_TO talk_VVI about_II the_AT voltage_NN1 between_II two_MC points_NN2 ._. 
It_PPH1 will_VM ,_, in_RR21 general_RR22 ,_, depend_VV0 on_II the_AT path_NN1 chosen_VVN ._. 
For_IF the_AT static_JJ case_NN1 the_AT line_NN1 integral_JJ of_IO the_AT electric_JJ field_NN1 disappears_VVZ when_CS taken_VVN over_II a_AT1 closed_JJ path_NN1 ._. 
For_IF the_AT time-varying_JJ case_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ ,_, in_RR21 general_RR22 ,_, different_JJ from_II zero_MC ._. 
We_PPIS2 shall_VM introduce_VVI now_RT the_AT notation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT so-called_JJ electromotive_JJ force_NN1 ._. 
It_PPH1 is_VBZ rather_RG unfortunate_JJ to_TO call_VVI it_PPH1 a_AT1 force_NN1 because_CS it_PPH1 is_VBZ n't_XX one_PN1 ._. 
At_II the_AT same_DA time_NNT1 one_PN1 can_VM have_VHI sympathy_NN1 with_IW those_DD2 who_PNQS devised_VVD the_AT term_NN1 because_CS it_PPH1 is_VBZ the_AT tangential_JJ force_NN1 per_II unit_NN1 charge_NN1 integrated_VVN over_II a_AT1 closed_JJ path_NN1 ._. 
In_II other_JJ words_NN2 it_PPH1 is_VBZ the_AT work_NN1 done_VDN by_II taking_VVG a_AT1 unit_NN1 charge_NN1 round_II a_AT1 circuit_NN1 ._. 
The_AT definition_NN1 ,_, eqn_NN1 (_( 4.17_MC )_) ,_, though_CS often_RR used_VVN ,_, is_VBZ regrettably_RR not_XX sufficiently_RR general_JJ ._. 
When_CS the_AT closed_JJ path_NN1 is_VBZ in_II a_AT1 wire_NN1 loop_NN1 and_CC the_AT loop_NN1 is_VBZ in_II motion_NN1 then_RT a_AT1 force_NN1 due_II21 to_II22 the_AT magnetic_JJ field_NN1 is_VBZ present_JJ as_RR21 well_RR22 ,_, giving_VVG rise_NN1 to_II a_AT1 finite_JJ amount_NN1 of_IO work_NN1 in_II the_AT same_DA manner_NN1 ._. 
The_AT more_RGR general_JJ definition_NN1 will_VM be_VBI discussed_VVN in_II Section_NN1 4.7_MC ._. 
For_RT41 the_RT42 time_RT43 being_RT44 we_PPIS2 shall_VM use_VVI eqn_NN1 (_( 4.17_MC )_) ,_, which_DDQ substituted_VVD into_II eqn_NN1 (_( 4.10_MC )_) will_VM yield_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, a_AT1 restricted_JJ form_NN1 of_IO Faraday_NP1 's_GE law_NN1 (_( for_IF its_APPGE general_JJ form_NN1 ,_, see_VV0 Section_NN1 4.7_MC )_) ._. 
This_DD1 is_VBZ valid_JJ for_IF the_AT case_NN1 when_CS the_AT flux_NN1 linking_VVG a_AT1 stationary_JJ circuit_NN1 varies_VVZ as_II a_AT1 function_NN1 of_IO time_NNT1 ._. 
The_AT dimension_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) is_VBZ that_DD1 of_IO voltage_NN1 ,_, so_CS it_PPH1 is_VBZ not_XX surprising_JJ that_CST many_DA2 people_NN refer_VV0 to_II it_PPH1 as_CSA voltage_NN1 or_CC induced_JJ voltage_NN1 ,_, a_AT1 usage_NN1 into_II which_DDQ I_PPIS1 often_RR lapse_VV0 myself_PPX1 ._. 
However_RR ,_, when_CS we_PPIS2 want_VV0 to_TO emphasize_VVI the_AT ability_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) to_TO put_VVI charges_NN2 into_II motion_NN1 ,_, it_PPH1 seems_VVZ preferable_JJ to_TO accord_VVI to_II it_PPH1 its_APPGE full_JJ title_NN1 ,_, or_CC at_RR21 least_RR22 its_APPGE popular_JJ abbreviation_NN1 in_II the_AT form_NN1 of_IO e.m.f_NNU ._. 
We_PPIS2 have_VH0 now_RT derived_VVN a_AT1 new_JJ law_NN1 for_IF slowly_RR varying_JJ phenomena_NN2 ._. 
What_DDQ can_VM we_PPIS2 say_VVI about_II the_AT laws_NN2 derived_VVN in_II the_AT last_MD chapter_NN1 ?_? 
Are_VBR Ampre_NP1 's_GE law_NN1 and_CC Biot-Savart_NP1 's_GE law_NN1 still_RR valid_JJ ?_? 
And_CC the_AT relationship_NN1 derived_VVN between_II the_AT current_JJ density_NN1 and_CC the_AT vector_NN1 potential_NN1 (_( eqn_NN1 (_( 3.15_MC )_) )_) ,_, is_VBZ that_DD1 still_RR valid_JJ ?_? 
Yes_UH ,_, all_DB of_IO them_PPHO2 are_VBR still_RR true_JJ as_II a_AT1 good_JJ approximation_NN1 as_CS31 long_CS32 as_CS33 the_AT inequality_NN1 (_( 4.1_MC )_) stands_VVZ ._. 
A_AT1 slowly_RR varying_JJ current_NN1 will_VM produce_VVI a_AT1 vector_NN1 potential_NN1 (_( or_CC a_AT1 magnetic_JJ field_NN1 )_) varying_VVG at_II the_AT same_DA rate._NNU 4.2_MC ._. 
The_AT electric_JJ field_NN1 due_II21 to_II22 a_AT1 varying_JJ magnetic_JJ field_NN1 Let_VV0 us_PPIO2 consider_VVI a_AT1 two-dimensional_JJ case_NN1 where_CS the_AT magnetic_JJ field_NN1 at_II a_AT1 given_JJ moment_NN1 is_VBZ constant_JJ within_II a_AT1 cylinder_NN1 of_IO radius_NN1 a_AT1 ,_, and_CC is_VBZ zero_MC outside_II this_DD1 cylinder_NN1 ._. 
We_PPIS2 shall_VM further_RRR assume_VVI a_AT1 sinusoidal_JJ time_NNT1 variation_NN1 so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) The_AT assumed_JJ magnetic_JJ field_NN1 is_VBZ independent_JJ both_RR of_IO z_ZZ1 and_CC of_IO the_AT azimuth_NN1 angle_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, hence_RR the_AT electric_JJ field_NN1 follows_VVZ the_AT same_DA pattern_NN1 ;_; it_PPH1 is_VBZ independent_JJ of_IO z_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) and_CC depends_VVZ on_II R_ZZ1 only_RR ._. 
Then_RT eqn_NN1 (_( 4.3_MC )_) yields_VVZ the_AT scalar_JJ differential_JJ equations_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 find_VV0 by_II inspection_NN1 that_CST the_AT solutions_NN2 are_VBR &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT constant_NN1 may_VM be_VBI determined_VVN from_II the_AT continuity_NN1 of_IO the_AT electric_JJ field_NN1 at_II R_ZZ1 =_FO a_AT1 yielding_JJ &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ quite_RG interesting_JJ ._. 
A_AT1 time-varying_JJ magnetic_JJ field_NN1 creates_VVZ an_AT1 electric_JJ field_NN1 as_CSA suggested_VVN by_II eqn_NN1 (_( 4.3_MC )_) ,_, but_CCB this_DD1 is_VBZ not_XX necessarily_RR a_AT1 local_JJ relationship_NN1 ._. 
The_AT magnetic_JJ field_NN1 may_VM be_VBI confined_VVN to_II a_AT1 certain_JJ part_NN1 of_IO space_NN1 but_CCB the_AT resulting_JJ electric_JJ field_NN1 will_VM pervade_VVI all_DB space_NN1 ._. 
What_DDQ can_VM we_PPIS2 say_VVI about_II the_AT line_NN1 integral_JJ of_IO the_AT electric_JJ field_NN1 ?_? 
It_PPH1 follows_VVZ from_II eqn_NN1 (_( 4.9_MC )_) that_CST it_PPH1 is_VBZ finite_JJ if_CS the_AT path_NN1 encloses_VVZ the_AT time-varying_JJ flux_NN1 but_CCB zero_VV0 otherwise_RR ._. 
How_RRQ does_VDZ this_DD1 appear_VVI in_II practice_NN1 ?_? 
What_DDQ happens_VVZ if_CS we_PPIS2 place_VV0 a_AT1 conducting_NN1 wire_NN1 into_II the_AT electric_JJ field_NN1 ?_? 
The_AT mobile_JJ charges_NN2 in_II the_AT wire_NN1 are_VBR not_XX the_AT least_RGT concerned_JJ about_II the_AT origin_NN1 of_IO the_AT electric_JJ field_NN1 ._. 
They_PPHS2 react_VV0 in_II the_AT same_DA way_NN1 whether_CSW the_AT electric_JJ field_NN1 is_VBZ due_II21 to_II22 static_JJ charges_NN2 or_CC to_II a_AT1 time-varying_JJ magnetic_JJ field_NN1 ;_; under_II the_AT force_NN1 qE_NN1 they_PPHS2 rearrange_VV0 themselves_PPX2 so_BCL21 as_BCL22 to_TO cancel_VVI the_AT electric_JJ field_NN1 inside_II the_AT conducting_NN1 material_NN1 as_CSA shown_VVN in_II Fig._NN1 4.1(a)_FO ._. 
There_EX will_VM now_RT be_VBI an_AT1 additional_JJ electric_JJ field_NN1 (_( say_VV0 Ec_NP1 )_) due_II21 to_II22 the_AT presence_NN1 of_IO these_DD2 charges_NN2 ._. 
Note_VV0 however_RR that_CST &lsqb;_( formula_NN1 &rsqb;_) ,_, hence_RR the_AT charges_NN2 do_VD0 not_XX interfere_VVI with_IW the_AT line_NN1 integral_JJ of_IO the_AT electric_JJ field_NN1 around_II a_AT1 closed_JJ path_NN1 ._. 
It_PPH1 is_VBZ still_RR true_JJ that_CST whether_CSW we_PPIS2 regard_VV0 E_ZZ1 as_II the_AT original_JJ field_NN1 (_( without_IW the_AT contribution_NN1 of_IO the_AT charges_NN2 )_) or_CC as_CS21 if_CS22 Ec_NP1 were_VBDR added_VVN to_II the_AT original_JJ field_NN1 ._. 
So_RR what_DDQ is_VBZ the_AT voltage_NN1 we_PPIS2 are_VBR going_VVGK to_TO measure_VVI ?_? 
Let_VV0 us_PPIO2 take_VVI an_AT1 ideal_JJ voltmeter_NN1 (_( one_PN1 that_CST draws_VVZ no_AT current_JJ )_) and_CC measure_VV0 the_AT voltage_NN1 across_II the_AT terminals_NN2 of_IO the_AT wire_NN1 ._. 
The_AT voltage_NN1 measured_VVD still_RR depends_VVZ on_II the_AT path_NN1 ._. 
We_PPIS2 measure_VV0 finite_JJ voltage_NN1 if_CS the_AT time-varying_JJ flux_NN1 is_VBZ enclosed_VVN (_( Fig._NN1 4.1(b)_FO )_) and_CC zero_NN1 voltage_NN1 if_CS no_AT flux_NN1 is_VBZ enclosed_VVN (_( Fig._NN1 4.1(c)_FO )_) ._. 
What_DDQ happens_VVZ if_CS we_PPIS2 have_VH0 N_NP1 loops_NN2 (_( Fig._NN1 4.2(a)_FO )_) round_II the_AT varying_JJ flux_NN1 ?_? 
Surely_RR ,_, we_PPIS2 will_VM have_VHI an_AT1 induced_JJ voltage_NN1 &lsqb;_( formula_NN1 &rsqb;_) in_II each_DD1 one_PN1 of_IO them_PPHO2 ._. 
If_CS we_PPIS2 connect_VV0 the_AT loops_NN2 then_RT the_AT total_JJ induced_JJ voltage_NN1 will_VM be_VBI the_AT algebraic_JJ sum_NN1 of_IO the_AT individual_JJ voltages_NN2 ._. 
If_CS we_PPIS2 make_VV0 up_RP a_AT1 helical_JJ coil_NN1 (_( Fig._NN1 4.2(b)_FO )_) the_AT wires_NN2 are_VBR going_VVG round_RP always_RR in_II the_AT same_DA direction_NN1 so_CS the_AT voltages_NN2 simply_RR add_VV0 and_CC we_PPIS2 may_VM rewrite_VVI eqn_NN1 (_( 4.13_MC )_) in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT flux_NN1 enclosed_VVN by_II each_DD1 turn_NN1 ._. 
Let_VV0 us_PPIO2 go_VVI now_RT one_MC1 step_NN1 further_RRR and_CC consider_VVI a_AT1 resistive_JJ wire_NN1 ring_NN1 (_( Fig._NN1 4.3(a)_FO )_) ._. 
The_AT force_NN1 on_II the_AT mobile_JJ charges_NN2 is_VBZ still_RR qE_JJ but_CCB now_RT the_AT charges_NN2 may_VM follow_VVI the_AT electric_JJ field_NN1 all_DB the_AT way_NN1 around_II the_AT ring_NN1 ._. 
A_AT1 current_JJ is_VBZ set_VVN up_RP corresponding_VVG to_II the_AT J_ZZ1 =_FO oE_NNU relationship_NN1 ._. 
Then_RT &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS R1_FO is_VBZ the_AT resistance_NN1 of_IO the_AT loop_NN1 ._. 
As_CSA may_VM be_VBI expected_VVN the_AT ohmic_JJ voltage_NN1 drop_NN1 in_II the_AT wire_NN1 will_VM be_VBI equal_JJ to_II the_AT induced_JJ voltage_NN1 ._. 
Let_VV0 us_PPIO2 next_MD evaluate_VVI the_AT line_NN1 integral_JJ of_IO the_AT electric_JJ field_NN1 over_II the_AT closed_JJ mathematical_JJ curve_NN1 shown_VVN in_II Fig._NN1 4.3(b)_FO ._. 
As_CS31 long_CS32 as_CS33 the_AT total_JJ time-varying_JJ magnetic_JJ flux_NN1 is_VBZ enclosed_VVN ,_, the_AT line_NN1 integral_JJ &lsqb;_( formula_NN1 &rsqb;_) will_VM remain_VVI the_AT same_DA as_CSA in_II the_AT previous_JJ example_NN1 ._. 
Note_VV0 however_RR that_CST the_AT electric_JJ field_NN1 is_VBZ different_JJ on_II Sections_NN2 1_MC1 and_CC 3_MC and_CC the_AT line_NN1 integrals_NN2 on_II Sections_NN2 2_MC and_CC 4_MC vanish_VV0 altogether_RR (_( because_CS the_AT path_NN1 is_VBZ perpendicular_JJ to_II the_AT electric_JJ field_NN1 )_) ._. 
Let_VV0 us_PPIO2 replace_VVI now_RT the_AT mathematical_JJ curve_NN1 by_II thin_JJ wire_NN1 ._. 
What_DDQ will_VM determine_VVI the_AT current_JJ flowing_JJ in_II the_AT wire_NN1 ?_? 
The_AT tangential_JJ component_NN1 of_IO the_AT electric_JJ field_NN1 along_II the_AT wire_NN1 ._. 
But_CCB then_RT we_PPIS2 obtain_VV0 the_AT answer_NN1 that_CST the_AT current_NN1 is_VBZ larger_JJR in_II Section_NN1 3_MC than_CSN in_II Section_NN1 1_MC1 ,_, and_CC no_AT current_JJ flows_NN2 in_II Sections_NN2 2_MC and_CC 4_MC ._. 
Is_VBZ that_DD1 possible_JJ ?_? 
According_II21 to_II22 eqn_NN1 (_( 4.2_MC )_) that_DD1 is_VBZ not_XX possible_JJ ._. 
Taking_VVG the_AT divergence_NN1 of_IO both_DB2 sides_NN2 of_IO eqn_NN1 (_( 4.2_MC )_) we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ may_VM be_VBI recognized_VVN as_II the_AT continuity_NN1 equation_NN1 for_IF an_AT1 incompressible_JJ fluid_NN1 ._. 
It_PPH1 means_VVZ in_II our_APPGE case_NN1 (_( the_AT same_DA that_CST commonsense_JJ would_VM suggest_VVI )_) that_CST the_AT current_JJ in_II the_AT loop_NN1 must_VM everywhere_RL be_VBI the_AT same_DA ._. 
It_PPH1 varies_VVZ of_RR21 course_RR22 as_II a_AT1 function_NN1 of_IO time_NNT1 but_CCB not_XX as_II a_AT1 function_NN1 of_IO the_AT spatial_JJ coordinates_NN2 ._. 
So_RR J_ZZ1 must_VM remain_VVI constant_JJ ,_, but_CCB it_PPH1 can_VM only_RR remain_VVI constant_JJ if_CS the_AT tangential_JJ component_NN1 of_IO the_AT electric_JJ field_NN1 is_VBZ constant_JJ as_RR21 well_RR22 ._. 
Hence_RR charges_NN2 must_VM appear_VVI (_( see_VV0 Fig._NN1 4.3(c)_FO )_) ,_, producing_VVG an_AT1 electric_JJ field_NN1 Ec._NP1 which_DDQ will_VM counteract_VVI the_AT induced_JJ electric_JJ field_NN1 Ei_NP1 so_CS21 that_CS22 the_AT resultant_JJ electric_JJ field_NN1 E_ZZ1 =_FO Ei_NP1 +_FO Ec_NP1 is_VBZ constant_JJ everywhere_RL along_II the_AT wire_NN1 ._. 
What_DDQ difference_NN1 will_VM it_PPH1 make_VVI if_CSW the_AT wire_NN1 loop_NN1 is_VBZ placed_VVN in_II the_AT magnetic_JJ field_NN1 as_CSA shown_VVN in_II Fig._NN1 4.4_MC ?_? 
There_EX will_VM be_VBI a_AT1 current_JJ produced_VVN by_II the_AT electric_JJ field_NN1 as_CSA before_RT ;_; the_AT new_JJ feature_NN1 is_VBZ the_AT appearance_NN1 of_IO a_AT1 qv_RR X_ZZ1 B_ZZ1 force_NN1 on_II each_DD1 charge_NN1 element_NN1 ._. 
Since_CS the_AT velocity_NN1 of_IO the_AT charge_NN1 carriers_NN2 is_VBZ in_II the_AT azimuthal_JJ direction_NN1 ,_, the_AT force_NN1 will_VM act_VVI radially_RR ._. 
It_PPH1 is_VBZ the_AT same_DA story_NN1 again_RT ._. 
The_AT charges_NN2 can_VM not_XX leave_VVI the_AT wire_NN1 so_CS they_PPHS2 will_VM accumulate_VVI at_II the_AT outer_JJ and_CC inner_JJ surface_NN1 of_IO the_AT ring_NN1 until_CS a_AT1 radial_JJ electric_JJ field_NN1 is_VBZ produced_VVN that_CST will_VM cancel_VVI the_AT force_NN1 due_II21 to_II22 the_AT magnetic_JJ field_NN1 ._. 
In_II conclusion_NN1 ,_, I_PPIS1 wish_VV0 to_TO emphasize_VVI that_CST all_DB the_AT charge_NN1 rearrangements_NN2 discussed_VVN in_II this_DD1 section_NN1 occur_VV0 very_RG fast_RR ,_, much_RR faster_RRR than_CSN the_AT period_NN1 of_IO oscillation_NN1 of_IO the_AT magnetic_JJ field_NN1 ._. 
Inductance_NN1 and_CC mutual_JJ inductance_NN1 We_PPIS2 shall_VM start_VVI again_RT with_IW a_AT1 resistive_JJ ring_NN1 but_CCB make_VV0 two_MC modifications_NN2 ._. 
The_AT externally_RR impressed_VVN voltage_NN1 need_VM not_XX come_VVI from_II a_AT1 time-varying_JJ magnetic_JJ field_NN1 as_CSA in_II the_AT previous_JJ section_NN1 ,_, it_PPH1 may_VM be_VBI produced_VVN by_II a_AT1 signal_NN1 generator_NN1 ._. 
Secondly_RR ,_, we_PPIS2 shall_VM take_VVI into_II account_NN1 the_AT voltage_NN1 induced_VVN by_II the_AT current_JJ flowing_JJ in_II the_AT ring_NN1 ._. 
If_CS you_PPY solve_VV0 the_AT field_NN1 problem_NN1 and_CC exercise_VV0 due_JJ care_NN1 in_II getting_VVG the_AT signs_NN2 right_RR ,_, you_PPY will_VM find_VVI that_CST the_AT induced_JJ voltage_NN1 is_VBZ such_II21 as_II22 to_TO oppose_VVI the_AT voltage_NN1 that_CST created_VVD it_PPH1 ._. 
For_IF this_DD1 reason_NN1 it_PPH1 is_VBZ often_RR called_VVN a_AT1 back_NN1 e.m.f_NNU ._. 
The_AT resulting_JJ equation_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
But_CCB according_II21 to_II22 Faraday_NP1 's_GE law_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, hence_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 shall_VM now_RT define_VVI self-inductance_NN1 by_II the_AT relationship_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Substituting_VVG the_AT above_JJ definition_NN1 into_II eqn_NN1 (_( 4.32_MC )_) and_CC changing_VVG from_II &lsqb;_( formula_NN1 &rsqb;_) to_II d/dt_FU we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, as_CSA known_VVN from_II circuit_NN1 theory_NN1 ._. 
Eqn_NN1 (_( 4.34_MC )_) is_VBZ suitable_JJ for_IF the_AT measurement_NN1 of_IO L._NP1 We_PPIS2 impose_VV0 a_AT1 sinusoidal_JJ V_ZZ1 ext_NN1 ,_, we_PPIS2 know_VV0 R_ZZ1 (_( from_II calculations_NN2 or_CC measurement_NN1 )_) ,_, we_PPIS2 measure_VV0 I_PPIS1 ,_, calculate_VV0 dI/dt_FU ,_, and_CC then_RT L_ZZ1 ,_, being_VBG the_AT only_JJ unknown_JJ ,_, is_VBZ determined_VVN ._. 
Can_VM we_PPIS2 calculate_VVI L_ZZ1 ?_? 
Not_XX easily_RR ._. 
Even_RR for_IF such_DA a_AT1 simple_JJ geometry_NN1 as_CSA the_AT ring_NN1 there_EX are_VBR lots_PN of_IO difficulties_NN2 ._. 
In_II Section_NN1 3.7_MC we_PPIS2 managed_VVD to_TO get_VVI simple_JJ expressions_NN2 for_IF the_AT magnetic_JJ field_NN1 far_RR away_II21 from_II22 the_AT ring_NN1 and_CC in_II the_AT vicinity_NN1 of_IO the_AT axis_NN1 ._. 
But_CCB in_BCL21 order_BCL22 to_TO calculate_VVI the_AT flux_NN1 across_II the_AT ring_NN1 we_PPIS2 need_VV0 to_TO know_VVI either_RR the_AT magnetic_JJ field_NN1 at_II every_AT1 point_NN1 in_II the_AT interior_NN1 of_IO the_AT ring_NN1 or_CC the_AT vector_NN1 potential_NN1 along_II the_AT perimeter_NN1 of_IO the_AT ring_NN1 ._. 
Eqn_NN1 (_( 3.56_MC )_) is_VBZ complicated_VVN enough_RR and_CC that_DD1 does_VDZ not_XX even_RR take_VVI account_NN1 of_IO the_AT finite_JJ diameter_NN1 of_IO the_AT wire_NN1 ._. 
So_RR the_AT problem_NN1 is_VBZ pretty_RG complicated_JJ ._. 
Reference_NN1 books_NN2 give_VV0 the_AT answer_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS d_ZZ1 is_VBZ the_AT diameter_NN1 of_IO the_AT wire_NN1 ._. 
This_DD1 looks_VVZ simple_JJ enough_RR but_CCB ,_, believe_VV0 me_PPIO1 ,_, a_AT1 lot_NN1 of_IO sweat_NN1 has_VHZ gone_VVN into_II producing_VVG that_DD1 formula_NN1 ._. 
Eqn_NN1 (_( 4.34_MC )_) is_VBZ of_RR21 course_RR22 true_JJ for_IF any_DD closed_JJ circuit_NN1 upon_II which_DDQ a_AT1 time-varying_JJ voltage_NN1 is_VBZ impressed_VVN ._. 
In_RR21 general_RR22 ,_, R_ZZ1 is_VBZ the_AT resistance_NN1 of_IO the_AT circuit_NN1 and_CC L_ZZ1 may_VM be_VBI obtained_VVN from_II eqn_NN1 (_( 4.33_MC )_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT flux_NN1 enclosed_VVN by_II the_AT circuit_NN1 ._. 
Can_VM one_PN1 determine_VVI the_AT inductance_NN1 of_IO any_DD of_IO the_AT practical_JJ configurations_NN2 with_IW relative_JJ ease_NN1 ?_? 
Yes_UH ,_, there_EX are_VBR a_AT1 few_DA2 ,_, e.g._REX a_AT1 two-wire_JJ transmission_NN1 line_NN1 which_DDQ is_VBZ nothing_PN1 else_RR but_CCB two_MC parallel_JJ wires_NN2 carrying_VVG opposite_JJ currents_NN2 ._. 
For_IF two_MC infinite_JJ line_NN1 currents_NN2 the_AT magnetic_JJ field_NN1 was_VBDZ given_VVN by_II eqn_NN1 (_( 3.53_MC )_) ._. 
We_PPIS2 get_VV0 the_AT flux_NN1 per_II unit_NN1 length_NN1 by_II integrating_VVG the_AT flux_NN1 density_NN1 over_II the_AT space_NN1 between_II the_AT wires_NN2 (_( assumed_VVN to_TO be_VBI infinitely_RR thin_JJ )_) ,_, as_CSA follows_VVZ :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
Unfortunately_RR ,_, the_AT above_JJ integral_JJ diverges_VVZ ._. 
What_DDQ 's_VBZ wrong_JJ ?_? 
Obviously_RR the_AT limits_NN2 :_: we_PPIS2 should_VM not_XX have_VHI taken_VVN the_AT wires_NN2 infinitely_RR thin_JJ ._. 
These_DD2 infinities_NN2 are_VBR our_APPGE best_JJT friends_NN2 and_CC worst_JJT enemies_NN2 ._. 
By_II taking_VVG things_NN2 infinitely_RR long_JJ and_CC infinitely_RR thin_JJ and_CC infinitely_RR something_PN1 else_RR ,_, we_PPIS2 can_VM arrive_VVI at_II simple_JJ formulae_NN2 ._. 
When_CS later_RRR we_PPIS2 want_VV0 to_TO use_VVI those_DD2 formulae_NN2 it_PPH1 turns_VVZ out_RP not_XX infrequently_RR that_CST we_PPIS2 have_VH0 some_DD divergent_JJ results_NN2 ._. 
In_II the_AT present_JJ case_NN1 the_AT remedy_NN1 is_VBZ clear_JJ and_CC easy_JJ ._. 
We_PPIS2 need_VV0 to_TO assume_VVI a_AT1 finite_JJ wire_NN1 diameter_NN1 ,_, and_CC we_PPIS2 can_VM do_VDI that_DD1 without_IW landing_VVG in_II a_AT1 sea_NN1 of_IO further_JJR complications_NN2 ._. 
This_DD1 is_VBZ because_CS wires_NN2 used_VVN in_II practice_NN1 (_( e.g._REX copper_NN1 )_) are_VBR non-magnetic_JJ :_: we_PPIS2 do_VD0 n't_XX need_VVI to_TO worry_VVI about_II boundary_NN1 conditions_NN2 at_RR21 all_RR22 ._. 
The_AT only_JJ difference_NN1 between_II copper_NN1 wire_NN1 and_CC air_NN1 is_VBZ that_CST the_AT former_DA carries_VVZ the_AT current_JJ ._. 
There_EX is_VBZ no_AT reason_NN1 now_RT for_IF the_AT current_JJ density_NN1 to_TO deviate_VVI from_II uniform_JJ distribution_NN1 over_II the_AT cross-section_NN1 ,_, so_RR eqn_NN1 (_( 3.53_MC )_) still_RR correctly_RR describes_VVZ the_AT magnetic_JJ field_NN1 between_II the_AT wires_NN2 ._. 
Consequently_RR ,_, the_AT flux_NN1 per_II unit_NN1 length_NN1 between_II the_AT wires_NN2 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS a_AT1 is_VBZ the_AT radius_NN1 of_IO the_AT wire_NN1 ._. 
Is_VBZ this_DD1 the_AT total_JJ flux_NN1 ?_? 
No_UH ,_, because_CS the_AT magnetic_JJ field_NN1 can_VM penetrate_VVI the_AT wires_NN2 ._. 
The_AT contribution_NN1 of_IO a_AT1 single_JJ wire_NN1 may_VM be_VBI determined_VVN with_II31 reference_II32 to_II33 Fig._NN1 4.5_MC ._. 
At_II a_AT1 radius_NN1 r_ZZ1 the_AT amount_NN1 of_IO current_JJ enclosed_JJ is_VBZ (_( r/a_FU )_) 2I_FO ,_, hence_RR the_AT magnetic_JJ field_NN1 from_II Ampre_NP1 's_GE law_NN1 is_VBZ yielding_VVG for_IF the_AT flux_NN1 per_II unit_NN1 length_NN1 inside_II the_AT material_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS ba_NN1 then_RT this_DD1 internal_JJ flux_NN1 may_VM be_VBI neglected_VVN giving_VVG the_AT often_RR quoted_VVN formula_NN1 for_IF the_AT inductance_NN1 per_II unit_NN1 length_NN1 of_IO a_AT1 two-wire_JJ transmission_NN1 line_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, but_CCB remember_VV0 there_EX is_VBZ an_AT1 internal_JJ flux_NN1 as_RR21 well_RR22 (_( and_CC a_AT1 corresponding_JJ internal_JJ inductance_NN1 )_) ,_, which_DDQ may_VM not_XX always_RR be_VBI negligible_JJ ._. 
Next_MD we_PPIS2 shall_VM look_VVI at_II a_AT1 long_JJ solenoid_NN1 filled_VVN with_IW a_AT1 high-permeability_JJ magnetic_JJ material_NN1 as_CSA shown_VVN in_II Fig._NN1 3.9_MC (_( p._NNU 73_MC )_) ._. 
Then_RT the_AT magnetic_JJ field_NN1 inside_II the_AT material_NN1 (_( from_II eqn_NN1 (_( 3.72_MC )_) )_) is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, uniform_NN1 across_II the_AT cross-section_NN1 (_( of_IO area_NN1 S0_FO )_) of_IO the_AT core_NN1 ._. 
Hence_RR the_AT magnetic_JJ flux_NN1 produced_VVN inside_II the_AT solenoid_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT voltage_NN1 induced_VVN in_II each_DD1 turn_NN1 of_IO the_AT solenoid_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) hence_RR the_AT self-inductance_NN1 comes_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Let_VV0 us_PPIO2 tap_VVI now_RT our_APPGE solenoid_NN1 at_II a_AT1 certain_JJ point_NN1 (_( Fig._NN1 4.6(a)_FO )_) so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 have_VH0 now_RT two_MC separate_JJ solenoids_NN2 with_IW N1_FO and_CC N2_FO turns_VVZ respectively_RR ._. 
How_RRQ could_VM we_PPIS2 determine_VVI their_APPGE inductances_NN2 ?_? 
Nothing_PN1 can_VM be_VBI simpler_JJR ,_, just_RR use_VV0 eqn_NN1 (_( 4.43_MC )_) above_RL and_CC get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
I_PPIS1 am_VBM afraid_JJ the_AT above_JJ formulae_NN2 are_VBR wrong_JJ because_CS our_APPGE new_JJ solenoids_NN2 do_VD0 n't_XX look_VVI the_AT same_DA as_CSA the_AT old_JJ one_PN1 ._. 
As_CSA shown_VVN in_II Figs_NN2 4.6(b)_FO and_CC (_( c_ZZ1 )_) the_AT new_JJ solenoids_NN2 have_VH0 a_AT1 core_NN1 of_IO length_NN1 l_ZZ1 ,_, although_CS the_AT wiring_NN1 is_VBZ confined_VVN to_II lengths_NN2 l1_FO and_CC l2_FO respectively_RR ._. 
Since_CS the_AT magnetic_JJ material_NN1 is_VBZ still_RR endowed_VVN with_IW the_AT property_NN1 of_IO concentrating_VVG the_AT field_NN1 lines_NN2 in_II itself_PPX1 ,_, the_AT flux_NN1 stays_VVZ constant_JJ for_IF the_AT whole_JJ length_NN1 of_IO the_AT core_NN1 ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS the_AT core_NN1 makes_VVZ up_RP a_AT1 closed_JJ circuit_NN1 as_CSA shown_VVN in_II Fig._NN1 4.7(a)_FO then_RT l_ZZ1 should_VM be_VBI taken_VVN as_II the_AT length_NN1 along_II the_AT dotted_JJ line_NN1 and_CC eqn_NN1 (_( 4.43_MC )_) is_VBZ still_RR applicable_JJ ._. 
In_II all_DB of_IO our_APPGE examples_NN2 so_RG far_RR we_PPIS2 could_VM calculate_VVI the_AT flux_NN1 through_II a_AT1 simple_JJ surface._NNU but_CCB how_RRQ should_VM we_PPIS2 take_VVI the_AT surface_NN1 when_CS there_EX is_VBZ no_AT magnetic_JJ material_NN1 to_TO guide_VVI the_AT field_NN1 lines_NN2 and_CC the_AT wire_NN1 itself_PPX1 is_VBZ of_IO a_AT1 complicated_JJ shape_NN1 ?_? 
Then_RT even_RR the_AT choice_NN1 of_IO the_AT surface_NN1 (_( having_VHG the_AT wire_NN1 as_CSA its_APPGE boundary_NN1 )_) might_VM tax_VVI to_II the_AT limit_NN1 one_PN1 's_GE meagre_JJ imagination_NN1 ,_, let_II21 alone_II22 the_AT calculation_NN1 of_IO the_AT flux_NN1 ._. 
The_AT practical_JJ answer_NN1 is_VBZ &quot;_" do_VD0 n't_XX calculate_VVI the_AT inductance_NN1 ,_, measure_VV0 it_PPH1 &quot;_" ,_, but_CCB if_CS for_IF some_DD reason_NN1 you_PPY must_VM do_VDI the_AT calculation_NN1 ,_, the_AT best_JJT approach_NN1 may_VM be_VBI to_TO abandon_VVI the_AT definition_NN1 of_IO eqn_NN1 (_( 4.33_MC )_) in_II31 favour_II32 of_II33 the_AT equivalent_JJ energetic_JJ definition_NN1 (_( see_VV0 Section_NN1 4.12_MC )_) &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 might_VM be_VBI simpler_JJR to_TO find_VVI the_AT total_JJ energy_NN1 in_II space_NN1 than_CSN the_AT flux_NN1 crossing_VVG a_AT1 given_JJ surface_NN1 ._. 
We_PPIS2 shall_VM now_RT go_VVI over_RP to_II the_AT definition_NN1 of_IO mutual_JJ inductance_NN1 with_IW the_AT aid_NN1 of_IO Fig._NN1 4.8_MC ._. 
The_AT current_JJ &lsqb;_( formula_NN1 &rsqb;_) flowing_VVG in_II loop_NN1 1_MC1 will_VM produce_VVI a_AT1 magnetic_JJ flux_NN1 &lsqb;_( formula_NN1 &rsqb;_) through_II loop_NN1 2_MC leading_JJ to_II the_AT definition_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
There_EX is_VBZ a_AT1 similar_JJ definition_NN1 for_IF M21_FO and_CC it_PPH1 can_VM be_VBI shown_VVN (_( see_VV0 Example_NN1 4.4_MC )_) that_CST M12_FO =M21_FO =M_FO ._. 
If_CS &lsqb;_( formula_NN1 &rsqb;_) is_VBZ time-varying_JJ ,_, the_AT voltage_NN1 induced_VVN in_II loop_NN1 2_MC is_VBZ M_ZZ1 (_( dI1_FO ,_, /dt_NN1 )_) ._. 
The_AT sign_NN1 of_IO M_ZZ1 depends_VVZ on_II the_AT convention_NN1 adopted_VVN for_IF the_AT direction_NN1 of_IO currents_NN2 in_II the_AT two_MC loops_NN2 ._. 
Circuit_NN1 engineers_NN2 take_VV0 the_AT mutual_JJ inductance_NN1 invariably_RR positive_JJ and_CC denote_VV0 the_AT direction_NN1 of_IO the_AT induced_JJ voltage_NN1 by_II a_AT1 dot_NN1 in_II the_AT circuit_NN1 diagram_NN1 ._. 
A_AT1 current_JJ flowing_JJ into_II the_AT inductance_NN1 L1_FO at_II the_AT dot_NN1 produces_VVZ a_AT1 voltage_NN1 V2_FO =_FO M(dI1/dt)_NN1 as_CSA shown_VVN in_II Fig._NN1 4.9_MC ._. 
Let_VV0 us_PPIO2 determine_VVI now_RT the_AT mutual_JJ inductance_NN1 between_II two_MC coils_NN2 wound_VVN on_II the_AT same_DA magnetic_JJ core_NN1 (_( Fig._NN1 4.7(b)_FO )_) ._. 
The_AT flux_NN1 produced_VVN by_II a_AT1 current_JJ in_II coil_NN1 1_MC1 is_VBZ given_VVN by_II eqn_NN1 (_( 4.42_MC )_) ,_, &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS the_AT subscript_NN1 1_MC1 referring_VVG to_TO coil_VVI 1_MC1 has_VHZ been_VBN attached_VVN to_II the_AT current_JJ and_CC number_NN1 of_IO turns_NN2 ._. 
Assuming_VVG now_CS21 that_CS22 all_DB the_AT flux_NN1 produced_VVN by_II coil_NN1 1_MC1 will_VM pass_VVI through_II coil_NN1 2_MC as_RR21 well_RR22 ,_, we_PPIS2 get_VV0 for_IF the_AT voltage_NN1 in_II coil_NN1 2_MC &lsqb;_( formula_NN1 &rsqb;_) ,_, whence_RRQ &lsqb;_( formula_NN1 &rsqb;_) ._. 
It_PPH1 is_VBZ interesting_JJ to_TO note_VVI that_CST &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS some_DD of_IO the_AT flux_NN1 produced_VVN by_II coil_NN1 1_MC1 does_VDZ not_XX pass_VVI through_II coil_NN1 2_MC (_( see_VV0 Fig._NN1 4.7(b)_FO )_) then_RT &lsqb;_( formula_NN1 &rsqb;_) appearing_VVG in_II eqn_NN1 (_( 4.50_MC )_) is_VBZ smaller_JJR leading_JJ to_II a_AT1 smaller_JJR value_NN1 of_IO mutual_JJ inductance_NN1 ._. 
Thus_RR when_CS some_DD flux_NN1 &quot;_" leaks_VVZ away_RL &quot;_" eqn_NN1 (_( 4.52_MC )_) becomes_VVZ an_AT1 inequality_NN1 :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
Alternatively_RR ,_, one_PN1 may_VM introduce_VVI a_AT1 coupling_NN1 coefficient_NN1 &lsqb;_( formula_NN1 &rsqb;_) and_CC rewrite_VV0 eqns_NN2 (_( 4.52_MC )_) and_CC (_( 4.53_MC )_) in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT above_JJ relationship_NN1 turns_VVZ out_RP to_TO be_VBI true_JJ for_IF any_DD two_MC magnetically_RR coupled_VVN circuits_NN2 ._. 
The_AT proof_NN1 will_VM be_VBI provided_VVN in_II Section_NN1 4.12_MC ._. 
Next_MD we_PPIS2 shall_VM determine_VVI the_AT mutual_JJ inductance_NN1 between_II two_MC concentric_JJ rings_NN2 at_II a_AT1 distance_NN1 h_ZZ1 apart_RL (_( Fig._NN1 3.17_MC ,_, p._NN1 82_MC )_) under_II the_AT assumption_NN1 that_CST the_AT upper_JJ ring_NN1 is_VBZ small_JJ ._. 
We_PPIS2 need_VV0 to_TO find_VVI the_AT magnetic_JJ flux_NN1 through_II ring_NN1 2_MC due_II21 to_II22 the_AT current_JJ in_II ring_NN1 1_MC1 ._. 
We_PPIS2 can_VM use_VVI the_AT formulae_NN2 derived_VVN in_II Section_NN1 3.7_MC for_IF the_AT vector_NN1 potential_NN1 and_CC magnetic_JJ field_NN1 in_II the_AT vicinity_NN1 of_IO the_AT axis_NN1 ._. 
With_IW the_AT aid_NN1 of_IO eqns_NN2 (_( 3.64_MC )_) and_CC (_( 3.65_MC )_) we_PPIS2 obtain_VV0 the_AT magnetic_JJ flux_NN1 in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) from_II the_AT vector_NN1 potential_NN1 ,_, and_CC in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) from_II the_AT magnetic_JJ field_NN1 ,_, giving_VVG of_RR21 course_RR22 identical_JJ results_NN2 ._. 
The_AT mutual_JJ inductance_NN1 is_VBZ then_RT &lsqb;_( formula_NN1 &rsqb;_) ._. 
Note_VV0 that_CST we_PPIS2 have_VH0 not_XX so_RG far_RR specified_VVD the_AT wire_NN1 thickness_NN1 of_IO either_DD1 ring_NN1 ._. 
For_IF the_AT mutual_JJ inductance_NN1 it_PPH1 is_VBZ not_XX needed_VVN ._. 
Finally_RR ,_, I_PPIS1 want_VV0 to_TO say_VVI a_AT1 few_DA2 words_NN2 about_II the_AT assumption_NN1 of_IO &quot;_" slowly_RR varying_JJ currents_NN2 &quot;_" in_II31 connection_II32 with_II33 the_AT last_MD example_NN1 ._. 
As_CSA you_PPY know_VV0 ,_, any_DD electromagnetic_JJ disturbance_NN1 (_( any_DD change_NN1 in_II anything_PN1 )_) propagates_VVZ with_IW the_AT velocity_NN1 of_IO light_NN1 ,_, something_PN1 we_PPIS2 have_VH0 so_RG far_RR neglected_VVN to_TO take_VVI into_II account_NN1 ._. 
When_CS calculating_VVG the_AT mutual_JJ inductance_NN1 we_PPIS2 assumed_VVD that_CST the_AT magnetic_JJ field_NN1 due_II21 to_II22 I1_FO appears_VVZ instantaneously_RR at_II the_AT second_MD ring_NN1 ._. 
In_II fact_NN1 it_PPH1 takes_VVZ time_NNT1 ,_, the_AT maximum_JJ time_NNT1 being_VBG equal_JJ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS l_ZZ1 is_VBZ the_AT largest_JJT possible_JJ distance_NN1 between_II any_DD two_MC points_NN2 in_II Fig._NN1 3.17_MC ._. 
If_CS the_AT current_JJ I1_FO may_VM be_VBI considered_VVN constant_JJ during_II the_AT time_NNT1 &lsqb;_( formula_NN1 &rsqb;_) then_RT we_PPIS2 are_VBR entitled_VVN to_TO talk_VVI about_II slow_JJ variation_NN1 ._. 
For_IF sinusoidal_JJ current_JJ variation_NN1 the_AT condition_NN1 is_VBZ that_CST the_AT period_NN1 of_IO oscillation_NN1 T_ZZ1 should_VM be_VBI much_RR larger_JJR than_CSN &lsqb;_( formula_NN1 &rsqb;_) ._. 
Using_VVG the_AT relationships_NN2 T_ZZ1 =_FO 1/f_FU and_CC c_ZZ1 =_FO &lsqb;_( formula_NN1 &rsqb;_) (_( where_CS f_ZZ1 and_CC &lsqb;_( formula_NN1 &rsqb;_) are_VBR the_AT frequency_NN1 and_CC wavelength_NN1 of_IO oscillations_NN2 )_) the_AT condition_NN1 may_VM be_VBI written_VVN in_II the_AT alternative_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ clear_JJ ._. 
As_CS31 long_CS32 as_CS33 the_AT maximum_JJ dimension_NN1 involved_VVN is_VBZ small_JJ in_II31 comparison_II32 with_II33 the_AT wavelength_NN1 ,_, we_PPIS2 are_VBR in_II the_AT slowly_RR varying_VVG region._NNU 4.4_MC ._. 
Kinetic_JJ inductance_NN1 We_PPIS2 have_VH0 defined_VVN inductance_NN1 by_II eqn_NN1 (_( 4.33_MC )_) without_IW discussing_VVG its_APPGE effect_NN1 upon_II the_AT current-voltage_JJ relationship_NN1 ._. 
Let_VV0 me_PPIO1 now_RT briefly_RR recall_VVI what_DDQ an_AT1 inductance_NN1 does_VDZ in_II a_AT1 circuit_NN1 ._. 
It_PPH1 delays_VVZ things_NN2 ._. 
If_CS we_PPIS2 apply_VV0 a_AT1 voltage_NN1 the_AT current_NN1 will_VM appear_VVI after_II some_DD delay_NN1 ._. 
If_CS we_PPIS2 switch_VV0 off_RP the_AT voltage_NN1 the_AT current_NN1 will_VM disappear_VVI after_II some_DD delay_NN1 ._. 
Is_VBZ there_EX anything_PN1 else_RR in_II an_AT1 electrical_JJ circuit_NN1 that_CST behaves_VVZ that_DD1 way_NN1 ?_? 
That_DD1 reminds_VVZ me_PPIO1 meeting_VVG an_AT1 American_JJ friend_NN1 of_IO mine_PPGE after_CS he_PPHS1 completed_VVD his_APPGE grand_JJ tour_NN1 of_IO Europe_NP1 ._. 
&quot;_" D'_VD0 you_PPY know_VVI &quot;_" ,_, he_PPHS1 told_VVD me_PPIO1 ,_, &quot;_" that_CST I_PPIS1 can_VM ask_VVI for_IF the_AT check_NN1 in_II seventeen_MC languages_NN2 ?_? 
&quot;_" &quot;_" Eighteen_MC &quot;_" ,_, I_PPIS1 said_VVD ,_, &quot;_" I_PPIS1 bet_VV0 you_PPY forgot_VVD to_TO include_VVI &quot;_" bill_NN1 &quot;_" ._. 
&quot;_" As_II the_AT above_JJ story_NN1 shows_VVZ it_PPH1 is_VBZ often_RR difficult_JJ to_TO think_VVI of_IO the_AT obvious_JJ ._. 
In_BCL21 order_BCL22 to_TO have_VHI a_AT1 current_JJ ,_, charge_NN1 carriers_NN2 must_VM be_VBI accelerated_VVN ,_, and_CC it_PPH1 takes_VVZ time_NNT1 to_TO accelerate_VVI particles_NN2 of_IO finite_JJ mass_NN1 ._. 
Hence_RR the_AT current_NN1 will_VM necessarily_RR lag_VVI behind_II the_AT voltage_NN1 causing_VVG its_APPGE rise_NN1 ._. 
Let_VV0 us_PPIO2 put_VVI now_RT the_AT relations_NN2 in_II mathematical_JJ form_NN1 ._. 
We_PPIS2 shall_VM write_VVI up_RP Newton_NP1 's_GE equation_NN1 for_IF the_AT case_NN1 when_CS the_AT force_NN1 is_VBZ provided_VVN by_II an_AT1 electric_JJ field_NN1 and_CC friction_NN1 is_VBZ present_JJ &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS k_ZZ1 is_VBZ a_AT1 constant_JJ characterizing_JJ friction_NN1 (_( to_TO be_VBI related_VVN presently_RR to_II more_RGR familiar_JJ constants_NN2 )_) ._. 
For_IF a_AT1 cylindrical_JJ piece_NN1 of_IO conducting_VVG material_NN1 of_IO length_NN1 l_ZZ1 and_CC cross-section_NN1 S0_FO &lsqb;_( formula_NN1 &rsqb;_) ,_, which_DDQ substituted_VVD into_II eqn_NN1 (_( 4.59_MC )_) leads_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 may_VM formally_RR write_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT kinetic_JJ resistance_NN1 ,_, and_CC &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT kinetic_JJ inductance_NN1 ._. 
If_CS you_PPY think_VV0 a_RR21 little_RR22 about_II it_PPH1 you_PPY will_VM be_VBI able_JK to_TO convince_VVI yourself_PPX1 that_CST Rk_NP1 is_VBZ nothing_PN1 other_II21 than_II22 the_AT old_JJ familiar_JJ resistance_NN1 (_( proportional_JJ to_II length_NN1 ;_; inversely_RR proportional_JJ to_II cross-section_NN1 )_) derived_VVD in_II a_AT1 different_JJ manner_NN1 ._. 
Whether_CSW one_PN1 defines_VVZ a_AT1 conductance_NN1 or_CC introduces_VVZ a_AT1 friction_NN1 term_NN1 they_PPHS2 are_VBR just_RR two_MC different_JJ ways_NN2 of_IO expressing_VVG the_AT empirical_JJ fact_NN1 that_CST the_AT electrons_NN2 '_GE velocity_NN1 does_VDZ not_XX go_VVI on_RP increasing_VVG indefinitely_RR in_II31 response_II32 to_II33 a_AT1 driving_JJ electric_JJ field_NN1 ._. 
Does_VDZ the_AT same_DA apply_VV0 to_II the_AT second_MD term_NN1 ?_? 
Is_VBZ that_DD1 also_RR a_AT1 different_JJ derivation_NN1 of_IO the_AT inductance_NN1 ?_? 
No_UH ,_, in_II deriving_VVG eqn_NN1 (_( 4.61_MC )_) we_PPIS2 have_VH0 not_XX talked_VVN about_II magnetic_JJ fields_NN2 and_CC induction_NN1 at_RR21 all_RR22 ._. 
The_AT kinetic_JJ inductance_NN1 owes_VVZ its_APPGE existence_NN1 to_II inertia_NN1 ._. 
Why_RRQ is_VBZ eqn_NN1 (_( 4.64_MC )_) so_RG little_RR known_VVN ?_? 
Because_CS under_II normal_JJ circumstances_NN2 this_DD1 kinetic_JJ inductance_NN1 is_VBZ negligible_JJ ._. 
Comparing_VVG eqns_NN2 (_( 4.63_MC )_) and_CC (_( 4.64_MC )_) you_PPY may_VM see_VVI that_CST Lk_NP1 =_FO Rk/k_FU ,_, and_CC if_CS you_PPY determine_VV0 k_ZZ1 from_II the_AT identity_NN1 &lsqb;_( formula_NN1 &rsqb;_) you_PPY will_VM find_VVI for_IF copper_NN1 that_CST &lsqb;_( formula_NN1 &rsqb;_) ._. 
So_RR you_PPY can_VM see_VVI that_CST the_AT time_NNT1 constant_NN1 involved_VVN is_VBZ very_RG small_JJ ._. 
The_AT kinetic_JJ inductance_NN1 does_VDZ ,_, however_RR ,_, acquire_VVI importance_NN1 in_II superconductors_NN2 ,_, where_CS the_AT resistance_NN1 disappears_VVZ altogether_RR ._. 
It_PPH1 may_VM be_VBI seen_VVN by_II comparing_VVG eqns_NN2 (_( 4.64_MC )_) and_CC (_( 4.35_MC )_) that_CST for_IF a_AT1 ring_NN1 made_VVN of_IO sufficiently_RR thin_JJ wire_NN1 the_AT kinetic_JJ inductance_NN1 may_VM exceed_VVI the_AT ordinary_JJ magnetic_JJ inductance_NN1 ._. 
And_CC this_DD1 may_VM very_RG well_RR occur_VVI in_II practice_NN1 because_CS superconductors_NN2 used_VVN in_II integrated_JJ circuits_NN2 can_VM have_VHI cross-sections_NN2 less_DAR than_CSN 10_MC -12_MC m_ZZ1 2_MC ._. 
It_PPH1 may_VM be_VBI said_VVN in_RR21 general_RR22 that_CST the_AT designer_NN1 of_IO a_AT1 superconducting_JJ circuit_NN1 needs_VVZ to_TO worry_VVI about_II the_AT response_NN1 time_NNT1 and_CC energy_NN1 of_IO superconducting_JJ electrons_NN2 ._. 
The_AT transformer_NN1 Let_VV0 us_PPIO2 return_VVI to_II the_AT configuration_NN1 of_IO two_MC coils_NN2 on_II a_AT1 magnetic_JJ core_NN1 (_( Fig._NN1 4.7_MC )_) and_CC apply_VV0 a_AT1 sinusoidal_JJ voltage_NN1 to_TO coil_VVI 1_MC1 from_II an_AT1 ideal_JJ voltage_NN1 generator_NN1 (_( having_VHG zero_MC internal_JJ resistance_NN1 )_) ._. 
In_II31 response_II32 to_II33 the_AT applied_JJ voltage_NN1 there_EX will_VM be_VBI a_AT1 current_JJ producing_VVG a_AT1 flux_NN1 that_CST will_VM induce_VVI voltages_NN2 both_RR in_II coils_NN2 1_MC1 and_CC 2_MC ._. 
If_CS the_AT coils_NN2 are_VBR lossless_JJ then_RT the_AT voltage_NN1 induced_VVN in_II coil_NN1 1_MC1 must_VM balance_VVI the_AT ,_, applied_JJ voltage_NN1 ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ._. 
Assuming_VVG for_IF the_AT moment_NN1 that_CST all_DB the_AT flux_NN1 is_VBZ contained_VVN within_II the_AT magnetic_JJ core_NN1 (_( so_CS21 that_CS22 the_AT amount_NN1 of_IO flux_NN1 crossing_NN1 coils_NN2 1_MC1 and_CC 2_MC is_VBZ the_AT same_DA )_) ,_, the_AT open_JJ circuit_NN1 voltage_NN1 in_II coil_NN1 2_MC is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, leading_VVG to_II the_AT familiar_JJ relationship_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Under_II open-circuit_JJ conditions_NN2 there_EX is_VBZ no_AT current_JJ flowing_JJ in_II coil_NN1 2_MC ._. 
The_AT current_JJ in_II coil_NN1 1_MC1 may_VM be_VBI obtained_VVN from_II the_AT relationship_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF an_AT1 applied_JJ voltage_NN1 of_IO &lsqb;_( formula_NN1 &rsqb;_) we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
What_DDQ happens_VVZ if_CS we_PPIS2 connect_VV0 a_AT1 resistance_NN1 RL_NP1 across_II the_AT terminals_NN2 of_IO coil_NN1 2_MC ?_? 
A_AT1 current_JJ I2_FO will_VM flow_VVI ._. 
But_CCB the_AT total_JJ flux_NN1 should_VM not_XX be_VBI affected_VVN because_CS it_PPH1 is_VBZ still_RR related_VVN to_II the_AT applied_JJ voltage_NN1 by_II eqn_NN1 (_( 4.66_MC )_) ._. 
Hence_RR an_AT1 additional_JJ current_JJ &lsqb;_( formula_NN1 &rsqb;_) will_VM be_VBI drawn_VVN from_II the_AT generator_NN1 so_BCL21 as_BCL22 to_TO satisfy_VVI the_AT equation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS L1_FO is_VBZ sufficiently_RR large_JJ so_CS21 that_CS22 &lsqb;_( formula_NN1 &rsqb;_) then_RT &lsqb;_( formula_NN1 &rsqb;_) may_VM be_VBI taken_VVN as_II the_AT total_JJ primary_JJ current_JJ I1_FO ,_, leading_VVG to_II the_AT relationship_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Eqns_NN2 (_( 4.68_MC )_) and_CC (_( 4.73_MC )_) are_VBR known_VVN as_II the_AT relationships_NN2 valid_JJ for_IF ideal_JJ transformers_NN2 ._. 
They_PPHS2 are_VBR certainly_RR very_RG useful_JJ for_IF an_AT1 engineer_NN1 because_CS they_PPHS2 relate_VV0 practical_JJ requirements_NN2 (_( e.g._REX how_RGQ large_JJ the_AT voltage_NN1 should_VM be_VBI at_II the_AT secondary_NN1 )_) to_TO design_VVI parameters_NN2 (_( what_DDQ the_AT turns_NN2 ratio_NN1 should_VM be_VBI )_) ._. 
How_RRQ are_VBR transformers_NN2 represented_VVN by_II circuit_NN1 engineers_NN2 ?_? 
Well_RR ,_, the_AT circuit_NN1 representation_NN1 shown_VVN in_II Fig._NN1 4.9_MC is_VBZ perfectly_RR adequate_JJ for_IF lossless_JJ transformers_NN2 ,_, but_CCB it_PPH1 would_VM not_XX yield_VVI the_AT equations_NN2 for_IF an_AT1 ideal_JJ transformer_NN1 ,_, not_XX even_RR for_IF the_AT case_NN1 of_IO perfect_JJ coupling_NN1 ,_, k_ZZ1 =_FO 1_MC1 ._. 
For_IF an_AT1 ideal_JJ transformer_NN1 we_PPIS2 need_VV0 to_TO introduce_VVI a_AT1 separate_JJ notation_NN1 which_DDQ we_PPIS2 shall_VM choose_VVI in_II the_AT form_NN1 shown_VVN in_II Fig._NN1 4.10_MC ._. 
The_AT relationship_NN1 between_II the_AT two_MC kinds_NN2 of_IO notations_NN2 is_VBZ not_XX an_AT1 obvious_JJ one_PN1 ._. 
It_PPH1 may_VM be_VBI shown_VVN (_( using_VVG the_AT criterion_NN1 that_CST they_PPHS2 lead_VV0 to_II the_AT same_DA set_NN1 of_IO equations_NN2 )_) that_CST by_II adding_VVG the_AT so-called_JJ leakage_NN1 inductances_NN2 to_II the_AT ideal_JJ transformer_NN1 the_AT two_MC representations_NN2 (_( Fig._NN1 4.11_MC )_) become_VV0 equivalent_JJ ._. 
One_PN1 may_VM add_VVI copper_NN1 losses_NN2 (_( resistive_JJ losses_NN2 in_II the_AT wires_NN2 )_) in_II a_AT1 similar_JJ manner_NN1 ._. 
If_CS you_PPY want_VV0 to_TO include_VVI iron_NN1 losses_NN2 (_( occurring_VVG in_II the_AT magnetic_JJ core_NN1 due_II21 to_II22 the_AT periodically_RR change_VV0 flux_NN1 )_) as_RR21 well_RR22 ,_, you_PPY should_VM better_RRR consult_VVI a_AT1 book_NN1 having_VHG a_AT1 bigger_JJR section_NN1 on_II transformers._NNU 4.6_MC Relative_JJ motion_NN1 of_IO conducting_VVG wire_NN1 and_CC magnetic_JJ field_NN1 Up_II21 to_II22 now_RT all_DB the_AT wires_NN2 have_VH0 been_VBN stationary_JJ ,_, and_CC the_AT magnetic_JJ field_NN1 has_VHZ varied_VVN as_II a_AT1 function_NN1 of_IO time_NNT1 ._. 
Let_VV0 us_PPIO2 see_VVI now_RT what_DDQ happens_VVZ when_RRQ the_AT wires_NN2 are_VBR made_VVN to_TO move_VVI with_IW uniform_JJ velocity_NN1 ._. 
As_CSA our_APPGE first_MD example_NN1 we_PPIS2 shall_VM take_VVI a_AT1 straight_JJ piece_NN1 of_IO wire_NN1 moving_VVG perpendicularly_RR to_II the_AT direction_NN1 of_IO a_AT1 static_JJ magnetic_JJ field_NN1 ._. 
As_II a_AT1 consequence_NN1 there_EX will_VM be_VBI a_AT1 force_NN1 qv_RR X_ZZ1 B_ZZ1 acting_VVG on_II the_AT charges_NN2 in_II the_AT wire_NN1 ;_; the_AT mobile_JJ charges_NN2 will_VM be_VBI displaced_VVN until_CS the_AT arising_VVG electric_JJ field_NN1 produces_VVZ an_AT1 equal_JJ and_CC opposite_JJ force_NN1 ._. 
Let_VV0 us_PPIO2 investigate_VVI now_RT the_AT converse_JJ problem_NN1 when_CS the_AT wire_NN1 is_VBZ stationary_JJ but_CCB the_AT equipment_NN1 producing_VVG the_AT magnetic_JJ field_NN1 is_VBZ moved_VVN lock_NN1 ,_, stock_NN1 ,_, and_CC barrel_NN1 with_IW the_AT same_DA velocity_NN1 in_II the_AT opposite_JJ direction_NN1 ._. 
One_PN1 's_GE first_MD impression_NN1 is_VBZ (_( or_CC should_VM be_VBI )_) that_CST only_RR relative_JJ motion_NN1 counts_NN2 ,_, hence_RR the_AT force_NN1 upon_II the_AT charges_NN2 should_VM be_VBI the_AT same_DA ._. 
Formally_RR ,_, we_PPIS2 would_VM have_VHI the_AT same_DA force_NN1 if_CS we_PPIS2 assumed_VVD (_( as_CSA many_DA2 textbooks_NN2 do_VD0 )_) that_CST a_AT1 magnetic_JJ field_NN1 moving_VVG with_IW a_AT1 velocity_NN1 in_RP gives_VVZ rise_NN1 to_II a_AT1 force_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ a_AT1 way_NN1 out_II21 of_II22 the_AT problem_NN1 but_CCB not_XX a_AT1 way_NN1 open_VV0 to_II us_PPIO2 ._. 
We_PPIS2 have_VH0 started_VVN by_II writing_VVG down_RP Maxwell_NP1 's_GE equations_NN2 and_CC claimed_VVD that_CST they_PPHS2 will_VM provide_VVI the_AT solution_NN1 to_II any_DD electromagnetic_JJ problem_NN1 ._. 
We_PPIS2 are_VBR not_XX entitled_VVN to_TO introduce_VVI a_AT1 new_JJ force_NN1 ._. 
According_II21 to_II22 our_APPGE equations_NN2 a_AT1 magnetic_JJ field_NN1 ,_, whether_CSW it_PPH1 moves_VVZ or_CC not_XX ,_, can_VM not_XX produce_VVI a_AT1 force_NN1 on_II a_AT1 stationary_JJ charge_NN1 ._. 
What_DDQ should_VM we_PPIS2 do_VDI ?_? 
Well_RR ,_, perhaps_RR the_AT trouble_NN1 is_VBZ that_CST we_PPIS2 wrote_VVD down_RP Maxwell_NP1 's_GE equations_NN2 in_II a_AT1 stationary_JJ frame_NN1 of_IO reference_NN1 ._. 
In_BCL21 order_BCL22 to_TO tackle_VVI the_AT present_JJ problem_NN1 we_PPIS2 should_VM change_VVI to_II a_AT1 coordinate_NN1 system_NN1 moving_VVG with_IW the_AT magnetic_JJ field_NN1 ._. 
This_DD1 is_VBZ certainly_RR a_AT1 possible_JJ approach_NN1 ,_, and_CC one_PN1 we_PPIS2 shall_VM adopt_VVI towards_II the_AT end_NN1 of_IO the_AT course_NN1 when_CS discussing_VVG relativity_NN1 ,_, but_CCB such_DA a_AT1 transformation_NN1 (_( though_CS convenient_JJ )_) should_VM not_XX be_VBI necessary_JJ ._. 
Maxwell_NP1 's_GE equations_NN2 written_VVN in_II the_AT stationary_JJ frame_NN1 of_IO reference_NN1 should_VM be_VBI perfectly_RR capable_JJ of_IO dealing_VVG with_IW the_AT problem_NN1 ._. 
Why_RRQ is_VBZ there_EX a_AT1 doubt_NN1 at_RR21 all_RR22 ?_? 
Some_DD doubts_NN2 may_VM arise_VVI by_II considering_VVG the_AT following_JJ problem_NN1 ._. 
Let_VV0 us_PPIO2 postulate_VVI the_AT existence_NN1 of_IO a_AT1 static_JJ magnetic_JJ field_NN1 that_CST is_VBZ constant_JJ over_II a_AT1 finite_JJ region_NN1 of_IO space_NN1 ,_, say_VV0 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC assume_VV0 that_CST this_DD1 magnetic_JJ field_NN1 is_VBZ bodily_RR moved_VVN in_II the_AT +x_FO direction_NN1 with_IW a_AT1 velocity_NN1 u_ZZ1 ._. 
What_DDQ will_VM happen_VVI to_II the_AT charges_NN2 in_II the_AT stationary_JJ wire_NN1 situated_VVN at_II (_( say_VV0 )_) x_ZZ1 =_FO 0_MC ?_? 
One_MC1 may_VM argue_VVI that_CST by_II moving_VVG the_AT magnetic_JJ field_NN1 nothing_PN1 has_VHZ changed_VVN at_II the_AT position_NN1 of_IO the_AT wire_NN1 ._. 
The_AT magnetic_JJ field_NN1 is_VBZ unchanged_JJ and_CC there_EX can_VM not_XX be_VBI an_AT1 electric_JJ field_NN1 either_RR because_CS a_AT1 constant_JJ magnetic_JJ field_NN1 is_VBZ unable_JK to_TO produce_VVI an_AT1 electric_JJ field_NN1 ._. 
The_AT above_JJ argument_NN1 is_VBZ wrong_JJ ;_; do_VD0 n't_XX let_VVI yourself_PPX1 be_VBI misled_VVN ._. 
Induction_NN1 is_VBZ not_XX a_AT1 local_JJ phenomenon_NN1 ._. 
Although_CS &lsqb;_( formula_NN1 &rsqb;_) ,_, it_PPH1 does_VDZ not_XX follow_VVI that_CST the_AT electric_JJ field_NN1 is_VBZ also_RR zero_MC at_II x_ZZ1 =_FO 0_MC ;_; the_AT curl_NN1 of_IO the_AT electric_JJ field_NN1 must_VM be_VBI zero_MC but_CCB not_XX the_AT electric_JJ field_NN1 itself_PPX1 ._. 
When_CS the_AT magnetic_JJ field_NN1 is_VBZ moved_VVN bodily_RR there_EX will_VM be_VBI certain_JJ places_NN2 in_II space_NN1 where_CS the_AT magnitude_NN1 of_IO the_AT magnetic_JJ field_NN1 is_VBZ changing_VVG (_( shaded_JJ areas_NN2 in_II Fig._NN1 4.12_MC )_) as_II a_AT1 function_NN1 of_IO time_NNT1 ,_, and_CC that_DD1 changing_JJ magnetic_JJ field_NN1 can_VM give_VVI rise_NN1 to_II an_AT1 electric_JJ field_NN1 at_II x_ZZ1 =_FO 0_MC ._. 
The_AT electric_JJ field_NN1 will_VM then_RT produce_VVI just_RR the_AT right_JJ force_NN1 for_IF displacing_VVG the_AT electrons_NN2 by_II the_AT right_JJ amount_NN1 ._. 
Let_VM21 's_VM22 attempt_VVI to_TO obtain_VVI a_AT1 general_JJ solution_NN1 of_IO this_DD1 problem_NN1 ._. 
We_PPIS2 shall_VM assume_VVI a_AT1 static_JJ magnetic_JJ field_NN1 &lsqb;_( formula_NN1 &rsqb;_) which_DDQ will_VM vary_VVI as_CSA &lsqb;_( formula_NN1 &rsqb;_) when_CS moved_VVN bodily_RR by_II a_AT1 velocity_NN1 u_ZZ1 ._. 
The_AT equation_NN1 to_TO solve_VVI is_VBZ eqn_NN1 (_( 4.3_MC )_) which_DDQ we_PPIS2 shall_VM write_VVI here_RL again_RT :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
By_II differentiating_JJ eqn_NN1 (_( 4.77_MC )_) we_PPIS2 get_VV0 &lsqb;_( formula_NN1 &rsqb;_) ._. 
But_CCB according_II21 to_II22 eqn_NN1 (_( A.7_FO )_) &lsqb;_( formula_NN1 &rsqb;_) solution_NN1 of_IO eqn_NN1 (_( 4.3_MC )_) may_VM be_VBI recognized_VVN as_CSA being_VBG given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
We_PPIS2 are_VBR now_RT entitled_VVN to_TO write_VVI the_AT force_NN1 on_II the_AT charges_NN2 as_CSA &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS the_AT magnetic_JJ field_NN1 is_VBZ moved_VVN with_IW a_AT1 velocity_NN1 -v_JJ the_AT force_NN1 on_II the_AT charges_NN2 is_VBZ the_AT same_DA as_CSA if_CS the_AT wire_NN1 moved_VVN with_IW a_AT1 velocity_NN1 v_ZZ1 ,_, and_CC we_PPIS2 have_VH0 managed_VVN to_TO prove_VVI all_DB that_CST from_II Maxwell_NP1 's_GE equations._NNU 4.7_MC ._. 
Faraday_NP1 's_GE law_NN1 In_BCL21 order_BCL22 to_TO derive_VVI the_AT general_JJ form_NN1 of_IO Faraday_NP1 's_GE law_NN1 we_PPIS2 need_VV0 to_TO notice_VVI that_DD1 motion_NN1 of_IO charge_NN1 along_II the_AT wire_NN1 may_VM be_VBI caused_VVN both_RR by_II electric_JJ and_CC magnetic_JJ fields_NN2 ._. 
Hence_RR for_IF the_AT calculation_NN1 of_IO e.m.f._NNU in_II a_AT1 moving_JJ loop_NN1 we_PPIS2 should_VM take_VVI both_DB2 effects_NN2 into_II account_NN1 leading_VVG to_II a_AT1 definition_NN1 in_II31 terms_II32 of_II33 the_AT force_NN1 per_II unit_NN1 charge_NN1 as_CSA follows_VVZ :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
Making_VVG use_NN1 of_IO eqn_NN1 (_( 4.79_MC )_) and_CC of_IO the_AT vector_NN1 derivative_NN1 relation_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, we_PPIS2 get_VV0 finally_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ the_AT general_JJ form_NN1 of_IO Faraday_NP1 's_GE law_NN1 where_CS the_AT derivative_NN1 is_VBZ now_RT taken_VVN in_II the_AT frame_NN1 moving_VVG with_IW the_AT wire._NNU 4.8_MC ._. 
Flux_NN1 cutting_NN1 Let_VV0 us_PPIO2 take_VVI a_AT1 loop_NN1 immersed_VVN in_II a_AT1 magnetic_JJ field_NN1 as_CSA shown_VVN in_II Fig._NN1 4.13(a)_FO and_CC assume_VV0 that_CST a_AT1 section_NN1 of_IO the_AT loop_NN1 moves_VVZ a_AT1 distance_NN1 dl_MC coming_VVG to_II a_AT1 position_NN1 shown_VVN in_II Fig._NN1 4.13(b)_FO after_CS a_AT1 time_NNT1 dt_NNU ._. 
This_DD1 elementary_JJ piece_NN1 of_IO wire_NN1 moving_VVG with_IW a_AT1 velocity_NN1 dl/dt_NN1 contributes_VVZ to_II the_AT e.m.f._NNU the_AT amount_NN1 which_DDQ may_VM be_VBI rewritten_VVN in_II the_AT form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT amount_NN1 of_IO flux_NN1 cut_VVN by_II the_AT moving_JJ part_NN1 of_IO the_AT loop_NN1 ._. 
If_CS several_DA2 distinct_JJ sections_NN2 of_IO the_AT conducting_NN1 loop_NN1 are_VBR in_II motion_NN1 then_RT the_AT flux_NN1 cut_VVN by_II each_DD1 section_NN1 needs_VVZ to_TO be_VBI summed_VVN algebraically_RR (_( each_DD1 contribution_NN1 taken_VVN as_CSA positive_JJ or_CC negative_JJ depending_JJ whether_CSW the_AT motion_NN1 of_IO that_DD1 section_NN1 increases_NN2 or_CC decreases_VVZ the_AT flux_NN1 linkage_NN1 )_) ._. 
The_AT conclusion_NN1 is_VBZ that_DD1 eqn_NN1 (_( 4.84_MC )_) is_VBZ valid_JJ under_II a_AT1 new_JJ set_NN1 of_IO conditions_NN2 ._. 
The_AT change_NN1 of_IO flux_NN1 may_VM be_VBI interpreted_VVN as_II the_AT amount_NN1 of_IO flux_NN1 cut_VVN by_II the_AT moving_JJ sections_NN2 of_IO the_AT loop._NNU 4.9_MC ._. 
Examples_NN2 on_II wires_NN2 moving_VVG in_II a_AT1 magnetic_JJ field_NN1 In_II the_AT following_JJ examples_NN2 we_PPIS2 shall_VM use_VVI Faraday_NP1 's_GE law_NN1 for_IF calculating_VVG the_AT e.m.f._NNU in_II moving_VVG loops_NN2 ._. 
Let_VV0 us_PPIO2 first_MD take_VVI a_AT1 static_JJ one-dimensional_JJ magnetic_JJ vector_NN1 field_NN1 pointing_VVG in_II the_AT z_ZZ1 direction_NN1 and_CC varying_VVG sinusoidally_RR in_II the_AT y_ZZ1 direction_NN1 as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS B0_FO and_CC k_ZZ1 are_VBR constants_NN2 ._. 
Assume_VV0 that_CST a_AT1 loop_NN1 lying_VVG in_II the_AT x_ZZ1 ,_, y_ZZ1 plane_NN1 (_( dimensions_NN2 shown_VVN in_II Fig._NN1 4.14_MC )_) moves_VVZ with_IW a_AT1 uniform_JJ velocity_NN1 v_ZZ1 in_II the_AT direction_NN1 of_IO the_AT positive_JJ y_ZZ1 axis_NN1 ._. 
For_IF simplicity_NN1 we_PPIS2 shall_VM further_RRR assume_VVI that_CST the_AT length_NN1 of_IO the_AT loop_NN1 in_II the_AT y_ZZ1 direction_NN1 is_VBZ smaller_JJR than_CSN the_AT period_NN1 of_IO the_AT magnetic_JJ field_NN1 ,_, &lsqb;_( formula_NN1 &rsqb;_) ._. 
If_CS the_AT rear_JJ section_NN1 of_IO the_AT loop_NN1 is_VBZ situated_VVN at_II y_ZZ1 =_FO -b/2_FU at_II time_NNT1 t_ZZ1 =_FO 0_MC ,_, then_RT the_AT positions_NN2 of_IO the_AT rear_NN1 and_CC front_JJ sections_NN2 will_VM be_VBI &lsqb;_( formula_NN1 &rsqb;_) at_II time_NNT1 t_ZZ1 ._. 
The_AT flux_NN1 enclosed_VVN is_VBZ then_RT &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT resulting_JJ e.m.f._NNU is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF our_APPGE second_MD example_NN1 we_PPIS2 shall_VM take_VVI a_AT1 magnetic_JJ field_NN1 varying_VVG sinusoidally_RR both_RR in_II space_NN1 and_CC in_II time_NNT1 :_: &lsqb;_( formula_NN1 &rsqb;_) ._. 
Assuming_VVG the_AT same_DA loop_NN1 travelling_VVG with_IW the_AT same_DA velocity_NN1 as_CSA in_II the_AT previous_JJ example_NN1 ,_, the_AT flux_NN1 through_II the_AT loop_NN1 varies_VVZ as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ._. 
Let_VV0 us_PPIO2 consider_VVI now_RT a_AT1 somewhat_RR different_JJ example_NN1 where_CS instead_II21 of_II22 uniform_JJ translation_NN1 the_AT loop_NN1 rotates_VVZ in_II a_AT1 constant_JJ magnetic_JJ field_NN1 ._. 
This_DD1 may_VM occur_VVI in_II the_AT same_DA physical_JJ configuration_NN1 as_CSA that_DD1 of_IO Fig._NN1 3.16_MC (_( p._NNU 81_MC )_) used_VVD for_IF calculating_VVG the_AT torque_NN1 upon_II a_AT1 single_JJ turn_NN1 of_IO current-carrying_JJ wire_NN1 ._. 
Assuming_VVG that_CST =_FO 0_MC at_II t_ZZ1 =_FO 0_MC the_AT angular_JJ position_NN1 of_IO the_AT loop_NN1 is_VBZ given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT flux_NN1 across_II the_AT loop_NN1 may_VM be_VBI obtained_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, whence_RRQ the_AT e.m.f._NNU is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Are_VBR you_PPY quite_RG happy_JJ with_IW this_DD1 result_NN1 ?_? 
I_PPIS1 suppose_VV0 you_PPY are_VBR ._. 
A_AT1 single_JJ turn_NN1 rotating_VVG in_II a_AT1 constant_JJ magnetic_JJ field_NN1 is_VBZ one_MC1 of_IO the_AT standard_JJ examples_NN2 of_IO the_AT application_NN1 of_IO Faraday_NP1 's_GE law_NN1 ._. 
The_AT result_NN1 is_VBZ used_VVN for_IF the_AT design_NN1 of_IO a.c._NN1 generators_NN2 so_CS it_PPH1 must_VM be_VBI all_RR right_JJ ._. 
Nevertheless_RR ,_, just_RR a_AT1 tiny_JJ little_JJ doubt_NN1 should_VM lurk_VVI somewhere_RL at_II the_AT back_NN1 of_IO your_APPGE mind_NN1 ._. 
In_II this_DD1 example_NN1 we_PPIS2 are_VBR not_XX concerned_JJ with_IW uniform_JJ translation_NN1 of_IO the_AT loop_NN1 but_CCB with_IW rotation_NN1 ._. 
The_AT loop_NN1 is_VBZ subjected_VVN to_II acceleration_NN1 ;_; there_EX is_VBZ a_AT1 centripetal_JJ force_NN1 acting_VVG upon_II the_AT electrons_NN2 inside_II the_AT wire_NN1 ._. 
The_AT present_JJ approach_NN1 is_VBZ permissible_JJ only_RR under_II the_AT condition_NN1 that_CST the_AT centripetal_JJ force_NN1 is_VBZ negligible_JJ in_II31 comparison_II32 with_II33 the_AT ev_NN1 X_ZZ1 B_ZZ1 force_NN1 ._. 
The_AT ratio_NN1 of_IO the_AT two_MC forces_NN2 may_VM be_VBI expressed_VVN as_CSA math_NN1 ;_; where_RRQ &lsqb;_( formula_NN1 &rsqb;_) is_VBZ the_AT so-called_JJ cyclotron_NN1 frequency_NN1 ._. 
Thus_RR effects_NN2 of_IO the_AT acceleration_NN1 are_VBR negligible_JJ as_CS31 long_CS32 as_CS33 &lsqb;_( formula_NN1 &rsqb;_) ._. 
In_II a_AT1 practical_JJ case_NN1 (_( say_VV0 )_) B_ZZ1 =_FO 1_MC1 T_ZZ1 and_CC the_AT loop_NN1 rotates_VVZ at_II 3000_MC revolution_NN1 per_II minute_NNT1 ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ,_, so_CS we_PPIS2 have_VH0 a_AT1 safe_JJ margin_NN1 ._. 
Nevertheless_RR ,_, keep_VV0 in_II mind_NN1 that_CST results_NN2 derived_VVN for_IF stationary_JJ or_CC uniformly_RR moving_VVG bodies_NN2 will_VM not_XX necessarily_RR apply_VVI when_RRQ the_AT body_NN1 is_VBZ accelerated_JJ ._. 
The_AT concept_NN1 of_IO flux_NN1 cutting_NN1 could_VM have_VHI equally_RR been_VBN used_VVN in_II the_AT calculations_NN2 involving_VVG static_JJ flux_NN1 ._. 
In_II our_APPGE first_MD example_NN1 where_CS the_AT magnetic_JJ field_NN1 varies_VVZ according_II21 to_II22 eqn_NN1 (_( 4.87_MC )_) the_AT flux_NN1 cut_VVN by_II the_AT front_JJ part_NN1 of_IO the_AT moving_JJ loop_NN1 in_II a_AT1 time_NNT1 dt_NNU is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC the_AT amount_NN1 cut_VVN by_II the_AT rear_JJ part_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR the_AT net_JJ flux_NN1 cut_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ,_, in_II agreement_NN1 with_IW eqn_NN1 (_( 4.90_MC )_) ._. 
4.10_MC ._. 
Eddy_VV0 currents_NN2 In_II this_DD1 section_NN1 the_AT effect_NN1 of_IO an_AT1 e.m.f._NNU on_II a_AT1 solid_JJ piece_NN1 of_IO conducting_VVG body_NN1 is_VBZ studied_VVN ._. 
In_II31 response_II32 to_II33 the_AT e.m.f._NNU a_AT1 current_JJ (_( an_AT1 eddy_NN1 current_JJ )_) flows_VVZ which_DDQ will_VM ,_, in_RR21 general_RR22 ,_, take_VV0 very_RG complicated_JJ paths_NN2 ._. 
We_PPIS2 shall_VM now_RT take_VVI a_AT1 very_RG simple_JJ case_NN1 when_CS a_AT1 conducting_NN1 disc_NN1 of_IO thickness_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, radius_NN1 b_ZZ1 ,_, and_CC of_IO conductivity_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ placed_VVN in_II the_AT magnetic_JJ field_NN1 of_IO eqn_NN1 (_( 4.19_MC )_) so_CS21 that_CS22 the_AT axes_NN2 coincide_VV0 ._. 
For_IF simplicity_NN1 we_PPIS2 shall_VM further_RRR assume_VVI that_CST b&lt;a_FO that_DD1 is_VBZ the_AT disc_NN1 is_VBZ completely_RR immersed_VVN into_II the_AT magnetic_JJ flux_NN1 ._. 
In_II the_AT absence_NN1 of_IO the_AT disc_NN1 the_AT electric_JJ field_NN1 is_VBZ given_VVN by_II eqn_NN1 (_( 4.22_MC )_) ._. 
Assuming_VVG that_CST the_AT magnetic_JJ field_NN1 produced_VVN by_II the_AT currents_NN2 in_II the_AT disc_NN1 is_VBZ negligible_JJ in_II31 comparison_II32 with_II33 the_AT magnetic_JJ field_NN1 postulated_VVD ,_, the_AT electric_JJ field_NN1 distribution_NN1 remains_VVZ the_AT same_DA and_CC the_AT corresponding_JJ current_JJ density_NN1 is_VBZ given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
At_II a_AT1 radius_NN1 R_ZZ1 the_AT current_JJ flows_NN2 in_II the_AT azimuthal_JJ direction_NN1 in_II a_AT1 tube_NN1 of_IO cross-section_NN1 &lsqb;_( formula_NN1 &rsqb;_) the_AT average_JJ power_NN1 dissipated_VVD by_II this_DD1 current_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) (_( cross-section_NN1 )_) 2_MC (_( resistance_NN1 of_IO the_AT tube_NN1 )_) &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT total_JJ power_NN1 dissipated_VVD in_II the_AT disc_NN1 may_VM then_RT be_VBI obtained_VVN by_II integration_NN1 as_CSA follows_VVZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
How_RRQ could_VM we_PPIS2 reduce_VVI the_AT eddy-current_JJ losses_NN2 in_II the_AT disc_NN1 ?_? 
First_MD ,_, we_PPIS2 have_VH0 to_TO choose_VVI a_AT1 low-conductivity_JJ material_NN1 ._. 
Secondly_RR ,_, we_PPIS2 could_VM interrupt_VVI the_AT currents_NN2 by_II insulating_JJ various_JJ parts_NN2 of_IO the_AT disc_NN1 from_II each_PPX221 other_PPX222 ._. 
These_DD2 problems_NN2 do_VD0 indeed_RR arise_VVI whenever_RRQV the_AT magnetic_JJ cores_NN2 of_IO coils_NN2 and_CC transformers_NN2 are_VBR made_VVN of_IO conducting_VVG materials_NN2 (_( mostly_RR iron_NN1 )_) ._. 
Both_DB2 remedies_NN2 suggested_VVN above_RL are_VBR put_VVN into_II practice_NN1 ._. 
The_AT iron_NN1 is_VBZ made_VVN into_II a_AT1 high-resistivity_JJ material_NN1 by_II adding_VVG silicon_NN1 ,_, and_CC its_APPGE cross-section_NN1 is_VBZ laminated_JJ as_CSA shown_VVN in_II Fig._NN1 4.15._MC 4.11_MC ._. 
Electromotive_JJ force_NN1 produced_VVN by_II rotating_VVG disc_NN1 We_PPIS2 shall_VM now_RT consider_VVI the_AT case_NN1 when_CS the_AT solid_JJ conducting_NN1 bodies_NN2 are_VBR in_II motion_NN1 ,_, the_AT simplest_JJT example_NN1 being_NN1 when_CS a_AT1 disc_NN1 is_VBZ rotated_VVN in_II a_AT1 constant_JJ magnetic_JJ field_NN1 as_CSA shown_VVN in_II Fig._NN1 4.16(a)_FO ._. 
There_EX will_VM now_RT be_VBI a_AT1 q_ZZ1 (_( v_ZZ1 X_ZZ1 B_ZZ1 )_) radial_JJ force_NN1 on_II the_AT charges_NN2 resulting_VVG in_II charge_NN1 separation_NN1 and_CC in_II the_AT appearance_NN1 of_IO an_AT1 electric_JJ field_NN1 ._. 
Let_VV0 us_PPIO2 complete_VVI now_RT the_AT circuit_NN1 with_IW the_AT aid_NN1 of_IO wires_NN2 and_CC brushes_NN2 (_( Fig._NN1 4.16(b)_FO )_) and_CC insert_VV0 an_AT1 ideal_JJ voltmeter_NN1 ._. 
What_DDQ is_VBZ the_AT voltage_NN1 measured_VVN ?_? 
It_PPH1 is_VBZ equal_JJ numerically_RR to_II the_AT e.m.f._NNU in_II the_AT closed_JJ circuit_NN1 ._. 
Since_CS the_AT wires_NN2 are_VBR stationary_JJ only_RR the_AT disc_NN1 contributes_VVZ to_II the_AT e.m.f._NNU ,_, hence_RR the_AT integration_NN1 over_II the_AT closed_JJ path_NN1 reduces_VVZ to_II integration_NN1 along_II the_AT radius_NN1 of_IO the_AT disc_NN1 ,_, yielding_VVG &lsqb;_( formula_NN1 &rsqb;_) ._. 
Can_VM we_PPIS2 work_VVI out_RP the_AT e.m.f._NNU with_IW the_AT aid_NN1 of_IO the_AT concepts_NN2 of_IO flux_NN1 linkage_NN1 or_CC flux_NN1 cutting_NN1 ?_? 
Not_XX easily_RR ._. 
If_CS we_PPIS2 choose_VV0 the_AT circuit_NN1 in_II the_AT form_NN1 shown_VVN in_II Fig._NN1 4.16(c)_FO there_EX is_VBZ neither_RR flux_NN1 linkage_NN1 nor_CC flux_NN1 cutting_NN1 ._. 
However_RR ,_, if_CS we_PPIS2 choose_VV0 our_APPGE loop_NN1 as_CSA shown_VVN in_II fig._NN1 4.16(d)_FO ,_, where_CS the_AT boundary_NN1 moves_VVZ with_IW the_AT rotating_JJ disc_NN1 ,_, the_AT both_RR flux_NN1 linkage_NN1 and_CC flux_NN1 cutting_NN1 make_VV0 sense_NN1 and_CC lead_VV0 to_II the_AT desired_JJ result_NN1 ._. 
Let_VV0 us_PPIO2 now_RT make_VVI the_AT problem_NN1 a_RR21 little_RR22 more_RGR complicated_JJ by_II assuming_VVG that_DD1 only_JJ part_NN1 of_IO the_AT disc_NN1 is_VBZ in_II an_AT1 appreciable_JJ magnetic_JJ field_NN1 ,_, as_CSA shown_VVN in_II Fig._NN1 4.17(a)_FO ._. 
What_DDQ happens_VVZ if_CS we_PPIS2 rotate_VV0 the_AT disc_NN1 ._. 
An_AT1 e.m.f._NNU will_VM duly_RR appear_VVI ._. 
Why_RRQ ?_? 
In_II the_AT region_NN1 permeated_VVD by_II the_AT magnetic_JJ field_NN1 the_AT charges_NN2 are_VBR subjected_VVN to_II radial_JJ forces_NN2 and_CC as_II a_AT1 consequence_NN1 eddy_NN1 currents_NN2 will_VM flow_VVI ._. 
The_AT resulting_JJ current_JJ distribution_NN1 would_VM be_VBI terribly_RR difficult_JJ to_TO calculate_VVI ,_, but_CCB one_PN1 can_VM make_VVI rough_JJ guess_NN1 and_CC claim_VVI that_CST it_PPH1 will_VM flow_VVI along_II the_AT dotted_JJ lines_NN2 shown_VVN in_II Fig._NN1 4.17(b)_FO ._. 
Omitting_VVG the_AT voltmeter_NN1 from_II our_APPGE circuit_NN1 ,_, so_CS21 that_CS22 a_AT1 current_JJ can_NN1 How_RRQ in_II the_AT resistive_JJ wire_NN1 ,_, it_PPH1 may_VM be_VBI seen_VVN that_CST some_DD of_IO the_AT current_NN1 will_VM be_VBI forced_VVN through_II the_AT wire_NN1 hence_RR the_AT device_NN1 works_VVZ as_II a_AT1 generator._NNU 4.12_MC ._. 
Energy_NN1 and_CC forces_VVZ The_AT force_NN1 acting_VVG on_II a_AT1 circuit_NN1 placed_VVN in_II a_AT1 magnetic_JJ field_NN1 is_VBZ given_VVN by_II eqn_NN1 (_( 3.90_MC )_) ._. 
If_CS we_PPIS2 have_VH0 (_( say_VV0 )_) two_MC circuits_NN2 then_RT the_AT forces_NN2 upon_II each_PPX221 other_PPX222 may_VM be_VBI calculated_VVN by_II determining_VVG first_MD the_AT magnetic_JJ field_NN1 over_II all_DB space_NN1 and_CC then_RT applying_VVG eqn_NN1 (_( 3.90_MC )_) ._. 
An_AT1 alternative_JJ method_NN1 is_VBZ to_TO derive_VVI first_MD the_AT energy_NN1 of_IO a_AT1 circuit_NN1 (_( or_CC circuits_NN2 )_) and_CC then_RT use_VV0 the_AT principle_NN1 of_IO virtual_JJ work_NN1 to_TO find_VVI the_AT force_NN1 ._. 
In_II this_DD1 section_NN1 we_PPIS2 shall_VM have_VHI a_AT1 brief_JJ look_NN1 at_II the_AT latter_DA method_NN1 ._. 
How_RRQ can_VM we_PPIS2 find_VVI the_AT energy_NN1 of_IO a_AT1 current-carrying_JJ circuit_NN1 ?_? 
The_AT answer_NN1 may_VM be_VBI easily_RR obtained_VVN with_IW the_AT aid_NN1 of_IO circuit_NN1 theory_NN1 ._. 
If_CS we_PPIS2 start_VV0 with_IW the_AT initial_JJ condition_NN1 that_CST I_ZZ1 =_FO O_ZZ1 at_II t_ZZ1 =_FO 0_MC and_CC at_II a_AT1 later_JJR time_NNT1 t_ZZ1 the_AT current_JJ rises_NN2 to_II I_PPIS1 then_RT the_AT work_NN1 done_VDN on_II the_AT inductor_NN1 is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ._. 
The_AT same_DA result_NN1 may_VM derived_VVN from_II field_NN1 theory_NN1 in_II the_AT following_JJ manner_NN1 ._. 
If_CS a_AT1 current_JJ element_NN1 I_PPIS1 ds_MC2 is_VBZ displaced_JJ parallel_NN1 to_II itself_PPX1 a_AT1 distance_NN1 dl_MC in_II a_AT1 magnetic_JJ flux_NN1 density_NN1 B_ZZ1 then_RT the_AT work_NN1 done_VDN is_VBZ &lsqb;_( formula_NN1 &rsqb;_) ,_, leading_VVG again_RT to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
For_IF two_MC coupled_JJ inductors_NN2 a_AT1 similar_JJ derivation_NN1 (_( either_RR by_II circuit_NN1 or_CC by_II field_NN1 theory_NN1 )_) yields_NN2 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Let_VV0 us_PPIO2 rewrite_VVI the_AT above_JJ equation_NN1 in_II the_AT following_JJ form_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Since_CS the_AT stored_JJ energy_NN1 must_VM always_RR be_VBI Positive_JJ for_IF any_DD pair_NN of_IO I1_FO and_CC I2_FO ,_, we_PPIS2 may_VM choose_VVI &lsqb;_( formula_NN1 &rsqb;_) ,_, whence_RRQ it_PPH1 follows_VVZ that_CST the_AT condition_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ valid_JJ for_IF any_DD two_MC coupled_JJ inductors_NN2 ._. 
Let_VV0 us_PPIO2 take_VVI now_RT two_MC rigid_JJ loops_NN2 in_II which_DDQ constant_JJ currents_NN2 I1_FO and_CC I2_FO flow_NN1 ._. 
The_AT total_JJ stored_JJ energy_NN1 is_VBZ given_VVN by_II eqn_NN1 (_( 4.108_MC )_) ._. 
What_DDQ is_VBZ the_AT change_NN1 in_II energy_NN1 if_CS we_PPIS2 move_VV0 one_MC1 of_IO the_AT loops_NN2 by_II a_AT1 distance_NN1 dx_MC ?_? 
The_AT self-inductances_NN2 will_VM stay_VVI constant_JJ (_( because_CS the_AT loops_NN2 are_VBR rigid_JJ )_) ,_, but_CCB the_AT mutual_JJ inductance_NN1 depends_VVZ on_II the_AT distance_NN1 between_II the_AT current_JJ elements_NN2 so_CS it_PPH1 will_VM change_VVI by_II an_AT1 amount_NN1 dM_NNU ._. 
Thus_RR the_AT change_NN1 in_II stored_JJ energy_NN1 comes_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ,_, whence_RRQ one_PN1 could_VM conclude_VVI that_CST &lsqb;_( formula_NN1 &rsqb;_) ._. 
This_DD1 is_VBZ ,_, of_RR21 course_RR22 ,_, wrong_JJ as_CSA we_PPIS2 could_VM have_VHI guessed_VVN on_II the_AT basis_NN1 of_IO our_APPGE experience_NN1 with_IW forces_NN2 acting_VVG upon_II capacitor_NN1 plates_NN2 ._. 
We_PPIS2 need_VV0 to_TO include_VVI the_AT work_NN1 done_VDN by_II the_AT sources_NN2 in_II keeping_VVG the_AT currents_NN2 constant_JJ ._. 
This_DD1 amounts_VVZ to_II &lsqb;_( formula_NN1 &rsqb;_) ._. 
Hence_RR &lsqb;_( formula_NN1 &rsqb;_) ._. 
As_II an_AT1 example_NN1 let_VV0 us_PPIO2 now_RT work_VVI out_RP the_AT forces_NN2 between_II two_MC current-carrying_JJ rings_NN2 with_IW the_AT aid_NN1 of_IO our_APPGE new_JJ formula_NN1 ._. 
The_AT mutual_JJ inductance_NN1 is_VBZ given_VVN by_II eqn_NN1 (_( 4.57_MC )_) ._. 
The_AT force_NN1 may_VM then_RT be_VBI calculated_VVN as_CSA &lsqb;_( formula_NN1 &rsqb;_) ,_, in_II agreement_NN1 with_IW eqn_NN1 (_( 3.93_MC )_) ._. 
Examples_NN2 4_MC 4.1_MC ._. 
A_AT1 circular_JJ coil_NN1 of_IO diameter_NN1 20_MC mm_NNU has_VHZ 100_MC turns_NN2 and_CC lies_VVZ at_II the_AT centre_NN1 of_IO a_AT1 solenoid_NN1 for_IF which_DDQ s_ZZ1 =_FO 0.3_MC m_NNO ,_, d_ZZ1 =_FO O.1_FO m_ZZ1 ,_, and_CC n_ZZ1 =_FO 800_MC ,_, the_AT coil_NN1 and_CC the_AT solenoid_NN1 being_VBG coaxial_JJ (_( s_ZZ1 =_FO total_JJ length_NN1 ,_, d_ZZ1 =_FO diameter_NN1 ,_, n_ZZ1 =_FO turns_VVZ per_II metre_NNU1 )_) ._. 
What_DDQ is_VBZ the_AT e.m.f._NNU induced_VVN in_II the_AT open-circuited_JJ coil_NN1 if_CS the_AT solenoid_NN1 is_VBZ fed_VVN with_IW a_AT1 current_JJ of_IO 3A_FO at_II 50_MC Hz_NNU ?_? 4.2_MC ._. 
Fig._NN1 4.18_MC shows_VVZ the_AT cross-section_NN1 of_IO a_AT1 straight_JJ transmission_NN1 line_NN1 carrying_VVG currents_NN2 &lsqb;_( formula_NN1 &rsqb;_) ,_, where_CS w_ZZ1 is_VBZ the_AT angular_JJ frequency_NN1 and_CC I0_FO is_VBZ a_AT1 constant_JJ ._. 
Derive_VV0 a_AT1 formula_NN1 for_IF the_AT e.m.f._NNU induced_VVN around_II a_AT1 loop_NN1 comprising_VVG a_AT1 length_NN1 of_IO l_NNU of_IO short-circuited_JJ cable_NN1 running_VVG parallel_RR with_IW the_AT transmission_NN1 line_NN1 ._. 
Assume_VV0 that_CST d&gt;a_FO and_CC ba_NN1 ._. 
Calculate_VV0 the_AT e.m.f._NNU when_CS I0_FO =_FO 1000_MC A_ZZ1 ,_, a_ZZ1 =_FO 1_MC1 m_ZZ1 ,_, b_ZZ1 =_FO 3_MC mm_NNU ,_, d_ZZ1 =_FO 5_MC m_NNO ,_, w_ZZ1 =_FO 100_MC &lsqb;_( formula_NN1 &rsqb;_) ,_, l_ZZ1 =_FO 1_MC1 m._NNO 4.3_MC ._. 
A_AT1 long_JJ ,_, thin-walled_NN1 ,_, non-magnetic_JJ ,_, conducting_VVG tube_NN1 of_IO radius_NN1 R_ZZ1 ,_, wall-thickness_NN1 &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC conductivity_NN1 &lsqb;_( formula_NN1 &rsqb;_) is_VBZ placed_VVN with_IW its_APPGE axis_NN1 parallel_NN1 to_II the_AT direction_NN1 of_IO a_AT1 uniform_JJ alternating_JJ magnetic_JJ field_NN1 &lsqb;_( formula_NN1 &rsqb;_) ._. 
Determine_VV0 the_AT current_JJ per_II unit_NN1 length_NN1 flowing_VVG in_II the_AT wall_NN1 ,_, the_AT magnetic_JJ field_NN1 inside_II the_AT tube_NN1 and_CC the_AT average_JJ power_NN1 dissipated_VVD per_II unit_NN1 length_NN1 ._. 
Neglect_VV0 edge_NN1 effects_NN2 that_CST is_VBZ assume_VV0 no_AT change_NN1 in_II the_AT variable_NN1 along_II the_AT tube._NNU 4.4_MC ._. 
Show_VV0 that_CST the_AT mutual_JJ inductance_NN1 between_II two_MC arbitrary_JJ coils_NN2 may_VM be_VBI written_VVN in_II the_AT form_NN1 (_( see_VV0 Fig._NN1 4.8_MC ,_, p._NN1 100_MC )_) math_NN1 ;_; ._. 
4.5_MC ._. 
&quot;_" Eqn_NN1 (_( 4.57_MC )_) gives_VVZ the_AT mutual_JJ inductance_NN1 of_IO two_MC concentric_JJ rings_NN2 a_AT1 distance_NN1 h_ZZ1 apart_II21 from_II22 each_PPX221 other_PPX222 ._. 
If_CS a1_FO and_CC a2_FO are_VBR interchanges_NN2 the_AT mutual_JJ inductance_NN1 will_VM not_XX remain_VVI invariant_JJ ,_, defying_VVG the_AT M12_FO =_FO M21_FO relationship_NN1 ._. 
What_DDQ is_VBZ the_AT cause_NN1 of_IO this_DD1 discrepancy_NN1 ?_? 4.6_MC ._. 
Determine_VV0 the_AT mutual_JJ impedance_NN1 between_II the_AT coil_NN1 and_CC the_AT solenoid_NN1 of_IO Example_NN1 4.1._MC 4.7_MC ._. 
Eqn_NN1 (_( 4.57_MC )_) gives_VVZ the_AT mutual_JJ inductance_NN1 of_IO two_MC concentric_JJ rings_NN2 a_AT1 distance_NN1 h_ZZ1 apart_II21 from_II22 each_PPX221 other_PPX222 ._. 
Assume_VV0 that_CST ring_VV0 1_MC1 carries_VVZ a_AT1 &lsqb;_( formula_NN1 &rsqb;_) and_CC the_AT resistance_NN1 and_CC self-inductance_NN1 of_IO ring_NN1 2_MC are_VBR given_VVN as_CSA L2_FO and_CC R2_FO ._. 
Show_VV0 that_CST the_AT mean_JJ value_NN1 of_IO the_AT force_NN1 F_ZZ1 between_II the_AT two_MC rings_NN2 (_( i.e._REX averaged_JJ over_II the_AT periodic_JJ time_NNT1 of_IO the_AT alternating_JJ current_JJ )_) is_VBZ given_VVN by_II &lsqb;_( formula_NN1 &rsqb;_) ,_, and_CC say_VV0 which_DDQ way_NN1 it_PPH1 acts_VVZ ._. 
It_PPH1 is_VBZ desired_VVN to_TO maximize_VVI the_AT ratio_NN1 (_( F/weight_FU of_IO ring_NN1 )_) ,_, and_CC for_IF this_DD1 purpose_NN1 it_PPH1 is_VBZ necessary_JJ to_TO decide_VVI whether_CSW for_IF any_DD given_JJ geometry_NN1 it_PPH1 is_VBZ better_JJR to_TO make_VVI the_AT ring_NN1 of_IO copper_NN1 or_CC of_IO aluminium_NN1 ._. 
Is_VBZ it_PPH1 possible_JJ to_TO give_VVI an_AT1 answer_NN1 without_IW having_VHG numerical_JJ values_NN2 for_IF anything_PN1 except_II the_AT properties_NN2 of_IO aluminium_NN1 and_CC copper_NN1 ?_? &lsqb;_( formula_NN1 &rsqb;_) ._. 
4.8_MC ._. 
Assume_VV0 an_AT1 axially_RR symmetric_JJ radial_JJ magnetic_JJ flux_NN1 density_NN1 in_II empty_JJ space_NN1 between_II the_AT radii_NN2 R1_FO and_CC R2_FO which_DDQ is_VBZ constant_JJ in_II the_AT vertical_JJ direction_NN1 (_( z_ZZ1 )_) ._. 
Place_VV0 now_RT a_AT1 thin_JJ wire_NN1 of_IO radius_NN1 &lsqb;_( formula_NN1 &rsqb;_) into_II the_AT magnetic_JJ field_NN1 at_II a_AT1 height_NN1 z_ZZ1 so_CS21 that_CS22 its_APPGE plane_NN1 is_VBZ perpendicular_JJ to_II the_AT Z-axis_NN1 ,_, and_CC each_DD1 element_NN1 of_IO the_AT wire_NN1 is_VBZ subjected_VVN to_II a_AT1 constant_JJ radial_JJ flux_NN1 density_NN1 Br_JJ ._. 
At_II t_ZZ1 =_FO 0_MC the_AT ring_NN1 is_VBZ released_VVN with_IW zero_MC initial_JJ velocity_NN1 ,_, (_( i_ZZ1 )_) Derive_VV0 the_AT equation_NN1 of_IO motion_NN1 for_IF the_AT ring_NN1 and_CC show_VV0 that_CST it_PPH1 is_VBZ independent_JJ of_IO the_AT cross-section_NN1 of_IO the_AT wire._NNU (_( ii_MC )_) Determine_VV0 the_AT position_NN1 and_CC velocity_NN1 of_IO the_AT ring_NN1 as_II a_AT1 function_NN1 of_IO time._NNU (_( iii_MC )_) What_DDQ is_VBZ the_AT limiting_JJ velocity_NN1 as_CSA t_ZZ1 </w>_NULL ?_? (_( iv_MC )_) In_II which_DDQ case_NN1 will_VM the_AT ring_NN1 fall_VVI faster_RRR ,_, if_CS it_PPH1 is_VBZ made_VVN of_IO copper_NN1 or_CC of_IO aluminium_NN1 ?_? 4.9_MC ._. 
A_AT1 transformer_NN1 supplied_VVN from_II a_AT1 220_MC V_NNU ,_, 50_MC Hz_NNU mains_NN1 has_VHZ an_AT1 iron_NN1 core_NN1 in_II which_DDQ the_AT peak_NN1 flux_NN1 density_NN1 ,_, 0.66_MC T_ZZ1 ,_, is_VBZ reached_VVN at_II a_AT1 magnetic_JJ field_NN1 of_IO 150_MC A_ZZ1 m-1_FO ._. 
The_AT cross-section_NN1 of_IO the_AT core_NN1 is_VBZ 1.5_MC 10_MC -3_MC m_ZZ1 2_MC and_CC its_APPGE volume_NN1 is_VBZ 9_MC 10_MC -4_MC m_ZZ1 3_MC ._. 
Determine_VV0 the_AT magnetization_NN1 current_JJ and_CC the_AT number_NN1 of_IO turns_NN2 in_II the_AT primary_JJ circuit._NNU 4.10_MC ._. 
A_AT1 rectangular_JJ loop_NN1 moving_VVG with_IW a_AT1 velocity_NN1 v_ZZ1 is_VBZ shown_VVN schematically_RR in_II Fig._NN1 4.14_MC (_( p._NNU 109_MC )_) and_CC the_AT resulting_JJ e.m.f._NNU is_VBZ given_VVN by_II eqn_NN1 (_( 4.90_MC )_) ._. 
Assume_VV0 now_CS21 that_CS22 the_AT loop_NN1 is_VBZ stationary_JJ and_CC the_AT magnetic_JJ field_NN1 (_( given_VVN by_II eqn_NN1 (_( 4.87_MC )_) )_) moves_VVZ with_IW the_AT same_DA velocity_NN1 in_II the_AT opposite_JJ direction_NN1 ._. 
Determine_VV0 the_AT e.m.f._NNU by_II taking_VVG the_AT line_NN1 integral_JJ of_IO the_AT electric_JJ field_NN1 around_II the_AT loop._NNU 4.11_MC ._. 
A_AT1 rectangular_JJ coil_NN1 is_VBZ located_VVN parallel_RR to_II a_AT1 long_JJ straight_JJ current-carrying_JJ wire_NN1 as_CSA shown_VVN in_II Fig._NN1 4.19_MC ._. 
Determine_VV0 the_AT e.m.f._NNU in_II the_AT coil_NN1 when_CS it_PPH1 is_VBZ rotated_VVN with_IW an_AT1 angular_JJ velocity_NN1 w._NNU 4.12_MC ._. 
Take_VV0 the_AT same_DA geometrical_JJ configuration_NN1 as_CSA in_II the_AT previous_JJ example_NN1 but_CCB assume_VV0 that_CST the_AT rectangular_JJ coil_NN1 is_VBZ moving_VVG with_IW a_AT1 constant_JJ velocity_NN1 u_ZZ1 in_II a_AT1 direction_NN1 perpendicular_NN1 to_II the_AT straight_JJ wire_NN1 ._. 
Determine_VV0 the_AT e.m.f._NNU in_II the_AT coil._NNU 4.13_MC ._. 
Find_VV0 the_AT torque_NN1 which_DDQ tends_VVZ to_TO align_VVI the_AT rotor_NN1 in_II the_AT arrangement_NN1 shown_VVN in_II Fig._NN1 4.20_MC with_IW 0_MC =_FO 45_MC and_CC 135_MC ,_, (_( i_ZZ1 )_) when_RRQ the_AT rotor_NN1 carries_VVZ no_AT current_JJ ,_, and_CC (_( ii_MC )_) when_CS it_PPH1 carries_VVZ a_AT1 direct_JJ current_NN1 of_IO 2_MC A._NNU The_AT rotor_NN1 inductance_NN1 is_VBZ 1_MC1 H_ZZ1 and_CC the_AT stator_NN1 inductance_NN1 has_VHZ maximum_JJ and_CC minimum_JJ values_NN2 1_MC1 H_ZZ1 and_CC 0.2_MC H._NP1 Assume_VV0 that_CST periodic_JJ inductances_NN2 vary_VV0 in_II a_AT1 sinusoidal_JJ manner_NN1 and_CC that_CST the_AT coils_NN2 are_VBR perfectly_RR coupled_VVN when_CS they_PPHS2 are_VBR in_II line_NN1 ._. 
