UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 24 Prof. Steven Errede LECTURE NOTES 24 MAXWELL S EQUATIONS

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1 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede LCTUR NOT 4 MAXWLL QUATION Thus far, we have the fllwing fur Maxwell equatins (in differential fr): ivergence and curl f bth and B specified nature f and B is fully defined i r TT ε ib r B r ( r) t B J r ρ ( r) ( Gauss' Law ) n agnetic nples { n agnetic charges} ( Faraday's Law ) ( r ) ( Apere's Law ) TT Hwever, there is a prble with this set f equatins r F r. Recall that i always fr an arbitrary vectr field, F( r) Apply this t Faraday s Law: B i( ) i ( i B) OK Apply this t Apere s Law: i i i ( B) ( JTT ) ( JTT ) Fr steady ttal currents: ( ) ρtt ij TT because Hwever, fr tie-varying situatins the cntinuity equatin (ttal charge cnservatin) ρtt i J TT BIG PROBLM!!! Let us investigate Apere s Law (in integral fr) fr the case f a parallel-plate capacitr: ( B) ida B d i C Jida Ienclsed i I enclsed C Bd Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

2 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede uppse we have an electric circuit cnsisting f sine wave functin generatr that i t V t V e ω and a parallel-plate capacitr: supplies/generates a tie dependent vltage Cplex ipedance f a capacitr: Cplex fr f Oh s law: V () t C n.b. V ( t ), Z C i* i + ω π f i iωc i* i I t Z I t and Z are cplex quantities. C Fr cntur lp shwn in figure: B d i I I c encl Get: I B( ρ ) as usual s n prble with this πρ What abut the fllwing cntur lp: Bd i c I encl!!! Cain t be true!!! We re issing sething! nergy flws acrss the gap between plates f parallel-plate capacitr - virtual phtns assciated with electric field f -plate capacitr! Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

3 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede Let s lk again at the cntinuity equatin: ρtt ( rt, ) i JTT But: ρtt ( rt, ) ε i rt (, ) (Gauss Law) ρtt ( rt, ) rt (, ) ε ( i ) ε i r: ijtt ε i If we cbine ε with JTT this cures the prble with Apere s Law: B ( rt, ) JTT ( rt, ) + ε JTT ( rt, ) + ε ( rt, ) ( B rt, ) ( JTT ( rt, )) ε i i + ( i rt (, )) ρtt ( rt, ) + ε ρ TT ( rt, ) ε ρtt ( rt, ) ρtt ( rt, ) +!!! Y IN!! Then: Nte the aesthetic/pleasing syetry: J r, t If TT ε new Apere s Law c B ( rt, ) (Faraday s Law), then B A changing electric field prduces a agnetic field! A changing agnetic field prduces a electric field! New ter is knwn as Maxwell s displaceent current: J r t ε (, ) New Apere s Law: B r t J r t + J r t (, ) TT (, ) (, ) J ε TT + Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved. 3

4 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede σ t Fr the -plate capacitr, the electric field () t ε Where: σ () t Q() t A where A area f ne plate σ () t Q() t Thus: () t ε ε A Thus: () Q() t t I t ε A ε A () Q t where I() t B ida Bid J ida J ida Then: ( ) TT + C encl Thus: Bid c ITT + ε ida s Fr -plate capacitr circuit with cntur C taken inside the gap f the -plate capacitr: Bd i c I enclsed TT + ε i da s B ingap ( ρ) ε ε A ()* I t πρ I t A ae answer!!! 4 Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

5 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede Thus the fur new Maxwell equatins are: i ρtt (Gauss Law) ε i B r, t (n agnetic nples/agnetic charges) B rt (, ) (Faraday s Law) B rt J rt + J rt (, ) TT (, ) (, ) J + ε Where: J (Apere s new Law) TT r t ε (, ) (Maxwell s displaceent curren Frce Law: F F + F q + qv B TT ρ Cntinuity equatin: J (charge cnservatin) ( rt, ) tt i TT can nw be derived fr Maxwell s eqns!! Nte that if agnetic charges g existed, then Maxwell s equatins wuld bece re syetrical : i ρtt (Gauss Law fr electric charges) ε i B ( rt, ) ρ TT ( rt, ) (Gauss Law fr agnetic charges) B ( rt, ) JTT (Faraday s Law) B JTT + ε (Apere s Law) TT c F r t q r t v r t B r t (, ) (, ) + (, ) (, ) FTT r t g B r t v r t r t c (, ) (, ) (, ) (, ) Please see/read Phy435 lecture ntes #8 fr re details abut agnetic nples. Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved. 5

6 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede In dielectric and agnetic edia we have: Maxwell s quatins in Matter lectric plarizatin Ρ electric diple ent per unit vlue. Magnetizatin Μ agnetic diple ent per unit vlue. Fr static fields: ρ ( r) Ρ i ( r) J ( r) Μ( r) Fr nn-static fields, a change in electric plarizatin ΔΡ has assciated with it a flw (i.e. a curren f charge a plarizatin current density JP ( r) which ust be included in the ttal current, I TT. Cnsider a tiny chunk/blck f plarized dielectric aterial The electric plarizatin Ρ induces surface charge densities σ Ρ i nˆ LH and + σ Ρ i n N ˆ RH N If the electric plarizatin Ρ increases an infinitesial aunt ΔΡ in tie Δ t, then the surface charges n each end als increase accrdingly, giving a net Plarizatin current: Δσ ΔΡ Δ IP da da i Δt Δt σ Ρ OR: dip da da J i P i da Ρ( rt, ) JP J (, ) P r t electric plarizatin current density (I units: Aps/ ) J r, t Μ r, t agnetic current density!!! Must nt be cnfused with: Because charge is charge is charge, there (f curse) exists a cntinuity equatin (expressing J r, t : electric charge cnservatin) fr electric plarizatin current density Ρ ρ ij P i ( i Ρ ) J P is essential t cnservatin f verall electric charge P 6 Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

7 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede Ttal electric charge density: ρ ( rt, ) ρ ( rt, ) + ρ ( rt, ) ρ ( rt, ) i Ρ( rt, ) Ttal electric current density: Then Gauss Law beces: TT JTT rt J rt J rt JP rt Ρ J + Μ + t (, ) (, ) + (, ) + (, ) i r t Tt r t r t + r t ε ε ρ ( rt, ) iρ( rt, ) ε (, ) ρ (, ) ρ (, ) ρ (, ) ε rt (, ) ε rt (, ) + Ρ ( rt, ) ( εrt i i i (, ) +Ρ( rt, )) r, t ρ r, t rt, ε rt, +Ρ rt, i and (New) Apere s Law (with Maxwell s displaceent current ter) B ( rt, ) J TT ( rt, ) + J ( rt, ) ( J (, ) (, ) r t + J r t + J ) + ε Ρ ( rt, ) rt (, ) J + Μ + + ε Ρ ( J + Μ ) + ε + ( J + Μ ) + ε +Ρ rt (, ) B ( J ) Μ + Then: H B Μ J + H J + H rt, Brt, Μ rt, and B ( rt, ) Faraday s Law: and i B (n agnetic nples) are unaffected by separatin f electric charge and electric current int and parts. Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved. 7

8 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede The fur Maxwell s quatins fr charges and currents nly: i ρ (Gauss' Law) ib (n agnetic charges) charges & currents nly B (Faraday's Law) t ) H J + (Apere's Law) The fur Maxwell equatins fr atter (i.e. dielectric and agnetic aterials) are: TT ε ε rt, ε rt, +Ρ rt, irt, ε irt, + i Ρ rt, ρ rt, rt rt Gauss Law: ρ ρ + i ρ ρ (, ) Ρ i (, ) Auxilliary Relatin: N agnetic nples: i B( r Faraday s Law:, B rt (, ) Apere s Law: B( rt, ) J ( rt, ) + J ( rt, ) With J ( TT ) ( J ( r J ( r JP ( r J ( r ), +, +, +, Ρ( rt, ) rt (, ) J + Μ + + ε ε Ρ rt, and J Μ and JP (, ) r t, H r t B r, t Μ r, t H J + Auxilliary Relatin: Fr linear dielectric and/r agnetic edia: Ρ rt, εχrt, e Μ ( rt, ) χh( rt, ) ( + e) Ke ε ε ( + χe) ( + χ) K ( + χ) ε H B ε ε χ 8 Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

9 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede Maxwell s quatins and Bundary Cnditins at Interfaces in Matter As we have seen previusly, in rder t btain relatins between nral (i.e. perpendicular) and tangential (i.e. parallel) cpnents f {,, Ρ} and/r { BH,, Μ} at an /ary between dielectric and/r agnetic edia, we ust use the integral fr(s) f Maxwell s equatins, because spatial derivatives f {,, Ρ} and/r { BH,, Μ} are nt defined at an /ary, fr Maxwell s equatins in differential fr: Gauss Law: i dτ ρtt dτ ( ρ ρ ) dτ v ε v v ε + enclsed enclsed enclsed ida QTT () t ( Q () t + Q () t ) ε ε enclsed ( r, t ) da Q ( t enclsed ) (, ) Ρ rtida Q t Auxiliary Relatin: r, t ε r, t +Ρ r, t ρbund rt rt (, ) Ρ i (, ) σ ( rt, ) ( rt, ) i nˆ Bund Ρ intf B r, t d B r, t ida N Magnetic Mnples: i τ Faraday s Law: v B d rt, da rt, d da Brt (, ) da i i C i dt i enclsed d dφm ( εfε() t rt (, ) d Brt (, ) da i C dt i dt Apere s Law: Brt, ida Brt, id J rt, + J rt, ida ( TT ) C B id I ( + I ( I t + I t + I t + I t C Auxiliary Relatin: encl encl encl encl () encl encl TT () P () () J Μ (, ) (, ) Μ(, ) JP (, ) K Μ nˆ r t H rt Brt rt Bund ρ rt rt intf Bund (, ) Μ i (, ) σ ( rt, ) ( rt, ) i nˆ P r t Μ enclsed enclsed d H da H d I ( + da I () t + i da i i i dt C (, ) intf Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved. 9

10 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede ) Apply the integral fr f Gauss Law at a dielectric /ary using infinitesially thin Gaussian pillbx extending slightly int dielectric aterial n either side f : ida Q Q + Q σ da + σ da enclsed enclsed enclsed TT ε ε ε ε ε Gives: ia i a σ a+ σa σtt a (at ) ε ε ε + (at ) ε ε r: σtt ( σ σ ) Here, the psitive directin is fr ediu () t ediu () enclsed da i Q σ da σ Likewise: enclsed Ρ ida Q σ da (at ) P P σ (at ) ince: V V V σ σ + σ ε TT ε (at ) Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

11 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede ince: ε ε V ε V V n ε σ (at ) ) iilarly, fr ibdτ B da v i (n agnetic nples), then at an : B ia B i a B B r: B B (at ) ince: H BΜ Bda i H+Μ ida r: Then: H a H a ( Μ aμ a) Then: B ( H +Μ) Hda i Μida i i i i (at ) H H Μ Μ σ Or: agnetic (at ) ffective agnetic charge at d dφ 3) Fr Faraday s Law: MF, ε d i ( Bda) C dt i dt between tw different edia, taking a clsed cntur C f width l extending slightly (i.e. infinitesially) int the aterial n either side f, as shwn : ide View: at an / ary F Ε {, r Ρ} r: { B, H r M} d i i B da dt i (in liit area f cntur lp, agnetic flux enclsed ) Thus: (at ) r: (at ) Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

12 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede ince: ε +Ρ And: ε Ρ Thus: ( i i) ( i i) ( i i ) Ρ Ρ In liit area f cntur lp agnetic flux enclsed Ρ Ρ (at ) Bd i I I C B i B i I + I encl encl 4) Finally, fr Apere s Law: ( + TOT ) Where encl encl TOT encl I TOT TOTAL current ( + + plarizatin) passing thrugh enclsing Aperian lp cntur C encl I J ida ε i da encl encl encl encl ITOT I + I + IP I encl J da Ρ da P P i i I encl J da da Μ i i N vlue current density JTOT, J, J r JP cntributes t encl I TOT in the liit area f cntur lp, hwever a surface current K, K, K Μ nˆ can cntribute! TOT In the liit that the enclsing Aperian lp cntur C shrinks t zer height /, the enclsed area f lp cntur, and encl d d Then: I ε da ε ΦΕ da i ε dt i dt ( Φ da Ε i enclsed flux f electric field lines) encl Ρ d d iilarly: I da ΦΡ Ρ da i Ρ dt i dt ( ΦΡ Ρ i da enclsed flux f electric plarizatin field lines) If ˆn is unit nral/perpendicular t (pinting fr ediu () int ediu (), nte that ( ˆn ) is nral/perpendicular t plane f the Aperian lp cntur. encl Thus: I ( ˆ ) ( ˆ TOT KTOT i n KTOT n) i Using: Ai( B C) Bi ( C A) encl I ( ˆ ) ( ˆ K i n K n) i Ci ( A B) encl I ( ˆ ) ( ˆ K i n K n) i { ( A B) i C} ITOT I + I KTOT K + K Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

13 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede In the liit that the enclsing Aperian lp cntur C (f width l) shrinks t zer height /, causing area f enclsed lp cntur, then: encl encl encl B i B i ITOT+ I ( ˆ ITOT KTOT n) i B ˆ ˆ B K TOT n K + K n (at ) ince: H BΜ and: B H +Μ then: ( B i B i) ( H i H i) + ( Μ i Μ i ) ( K ˆ) ( ˆ n + K n) (at ) We als see that: H ˆ H K n K (at ) and: Μ ˆ Μ n (at ) - cpnents f B are discntinuus at by ˆ KTOT n - cpnents f H are discntinuus at by K ˆ n - cpnents f Μ are discntinuus at by K nˆ If B A where A is the agnetic vectr ptential, then: TOT A A B B K nˆ K (at ) is equivalent t: TOT Fr linear agnetic edia: B H r: H B (at ) Then: H ˆ H K n (at ) is equivalent t: A A K n (at ) Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved. 3

14 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede uary f Maxwell s quatins In ifferential and Integral Frs rt, ependence) (uppress xplicit ε ε i ρ Gauss Law: ρ TOT ( ρ + ρ ) Use f the auxiliary relatin: Yields: i ε i+ i Ρ ρ ε +Ρ i Ρ N agnetic nples: i B B Faraday s Law: t Apere s Law: B ( JTOT + J) Use f auxiliary relatin: Yields: H J + t H BΜ JTOT J + J + J Ρ with J ε encl encl encl Gauss Law: idτ da Qtt Q + Q ΦΕ v i ε ε (lectric Flux) Use f the auxiliary relatin: ε +Ρ encl Yields: idτ da Q v i Φ (lectric isplaceent Flux) encl Ρ i dτ Ρ da Q v i ΦΡ (lectric Plarizatin Flux) N agnetic nples: ibdτ Bida v Faraday s Law: d dφ MF ε da i d Bda i C dt i dt Apere s Law: ( ib) ida ( TOT ) Bid J J da C + i encl encl encl encl encl encl ( ITOT + I ) ( I + I + IΡ + I ) Use f auxiliary relatin(s): H BΜ and ε +Ρ Yields: encl d ( H) da H d I + i da i C dt i encl 4 Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

15 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede uary f General Bundary Cnditins Obtained fr Integral Fr(s) f Maxwell s quatins uppressing explicit ( rt, ) dependence and defining ˆn (unit nral) pinting fr ediu () int ediu () as shwn in the figure : ide View: F {, r Ρ} r: F { B, H, r Μ } Then at the : + ε +Ρ ε ε Gauss Law: σ TOT ( σ σ ) σ Ρ Ρ σ V V σ σ + σ ε ε ε V V n ε σ TOT N Magnetic Mnples: B B H H Μ Μ σ H BΜ Faraday s Law: Ρ Ρ B ˆ ˆ B K n K + K n B Apere s Law: TOT ( ) A A K ˆ H H K n TOT A A K A Μ Μ ˆ K n agnetic Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved. 5

16 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede BC s pecific t Linear Hgeneus Istrpic ielectric and/r Magnetic Media: Ρεχ Μ χ H e ε ε +Ρ ε ε ( + χe) ( + χ) B H r H B ε Ke + χe K ( + χ) H BΜ ε The ary cnditins at the between linear dielectric r agnetic edia bece: ε ε V σ ε ε σ + σ Gauss Law: σ TOT ( σ σ + ) V TOT ( ) σ ε ε σ Ρ Ρ σ ε χ χ σ ε V V n ε σ e e N agnetic nples: B B H H M M σ H H B B agnetic Faraday s Law: ε ε Ρ Ρ ε ε ε χ χ e e Apere s Law: B ˆ ˆ B K TOT n K + K n H H A A K using B A TOT 6 Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved.

17 UIUC Physics 435 M Fields & urces I Fall eester, 7 Lecture Ntes 4 Prf. teven rrede H H K nˆ B B M ˆ M K n and χ H χ H χ χ B B A A K using B A Prfessr teven rrede, epartent f Physics, University f Illinis at Urbana-Chapaign, Illinis 5-8. All Rights Reserved. 7