6.641, Electromagnetic Fields, Forces, and Motion Prof. Markus Zahn Lecture 7: Polarization and Conduction

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1 6.641, Electromgnetic Fiels, Forces, n Motion Prof. Mrkus Zhn Lecture 7: Polriztion n Conuction I. Experimentl Oservtion A. Fixe oltge - Switch Close ( v= o ) As n insulting mteril enters free-spce cpcitor t constnt voltge more chrge flows onto the electroes; i.e. s x increses, i increses. B. Fixe Chrge - Switch open (i=0) As n insulting mteril enters free spce cpcitor t constnt chrge, the voltge ecreses; i.e. s x increses, v ecreses. II. Dipole Moel of Polriztion A. Polriztion ector P = Np = Nq ( p = q ipole moment) N ipoles/olume ( P is ipole ensity) q q Prof. Mrkus Zhn Pge 1 of 7

2 Courtesy of Krieger Pulishing. Use with permission. Prof. Mrkus Zhn Pge of 7

3 Q = qni = ρ insie S P pire chrge or equivlently polriztion chrge ensity Q = P i = ip = ρ insie S P (Divergence Theorem) P=qN i P = ρ P B. Guss Lw ( E) i = ρ = ρ ρ = ρ ip o totl u P u unpire chrge ensity; lso clle free chrge ensity ( E P ) = ρ i o u D = E P Displcement Flux Density o i D = ρ u C. Bounry Conitions Prof. Mrkus Zhn Pge 3 of 7

4 u u su S i D= ρ D i = ρ n i D D = σ P P sp S i P= ρ P i = ρ n i P P = σ ( ) i E = ρ ρ E i = ρ ρ n i E E = σ σ o u P o u P o su sp S D. Polriztion Current Density Q = qn = qni = P i [Amount of Chrge pssing through surfce re element ] i p Q P = = i t t [Current pssing through surfce re element ] = Jp i polriztion current ensity J p P = t Ampere s lw: xh = J J u p o E t P = Ju t o E t ( o ) = Ju E P t D t = Ju Prof. Mrkus Zhn Pge 4 of 7

5 III. Equipotentil Sphere in Uniform Electric Fiel r o o o lim Φ r, θ =Ercosθ Φ= E z =Ercosθ Φ ( r = R, θ ) = 0 3 R Φ ( r, θ ) = Eo r cos θ r This solution is compose of the superposition of uniform electric fiel plus the fiel ue to point electric ipole t the center of the sphere: p cosθ Φ = π ipole 4 or 3 with p = 4π E R o o This ipole is ue to the surfce chrge istriution on the sphere. 3 Φ R σ s ( r = R, θ ) = oer ( r = R, θ ) = o = oeo 1 cos 3 θ r r= R r r= R = 3 E cosθ o o Prof. Mrkus Zhn Pge 5 of 7

6 I. Artificil Dielectric v v E =, s E σ = = A q =σ sa = v C = q = A v _ υ E Courtesy of Hermnn A. Hus n Jmes R. Melcher. Use with permission. For sphericl rry of non-intercting spheres (s >> R) _ 3 3 P=4π R E iz P =N p =4π R E N o o z z o o N= 1 s 3 R 3 R 3 P= o 4π E= ψe o E ψe =4π s s ψ e (electric susceptiility) D= o E P= o 1 ψe E= E r (reltive ielectric constnt) 3 R = r o = o 1 ψ e = o 1 4π s Prof. Mrkus Zhn Pge 6 of 7

7 . Demonstrtion: Artificil Dielectric Courtesy of Hermnn A. Hus n Jmes R. Melcher. Use with permission. Prof. Mrkus Zhn Pge 7 of 7

8 Courtesy of Hermnn A. Hus n Jmes R. Melcher. Use with permission. v E= σs= E= v A q A q= σsa= v C= = v v i= ω C = R ( ) 3 o A R A C= =4πo s o s R=1.87 cm, s=8 cm, A= (0.4) m, =0.15m ω =π(50 Hz), R s =100 k Ω, =566 volts pek C=1.5 pf v 0 = ω CRs =(π) (50) (1.5 x 10-1 ) (10 5 ) 566 = volts pek Prof. Mrkus Zhn Pge 8 of 7

9 I. Plsm Conuction Moel (Clssicl) v m = q E m v p ν t n v p m = qe mνv t n p = n kt, p = nkt k=1.38 x 10-3 joules/ o K Boltzmnn Constnt A. Lonon Moel of Superconuctivity [ T 0, ν± 0] v v m = q E, m = qe t t J = q n v, J = q n v ( ) J qe v q n = ( q n v ) = q n = q n = E t t t m m ω p ( ) J qe v q n = ( q n v ) = q n = q n = E t t t m m ω p ω q n q n p =, ω p = m m (ω p = plsm frequency) For electrons: q - =1.6 x Couloms, m - =9.1 x kg n - =10 0 /m 3 1, = x10 frs/m o ω p ωp 10 f p = 9x10 π q n m 11 = 5.6 x10 Hz r/s Prof. Mrkus Zhn Pge 9 of 7

10 B. Drift-Diffusion Conuction [Neglect inerti] 0 v ( n k T) q k T m = q E m v v = E n ν t n m ν m ν n 0 v ( n k T) q k T m = q E m v v = E n ν t n mν mνn q n q k T J = q n v = E n m ν m ν q n q k T J = q n v = E n mν mν ρ =q n, ρ = q n J = ρ µ ED ρ J = ρ µ E D ρ µ q =, D = kt m ν m ν µ q kt =, D = mν m ν chrge moilities Moleculr Diffusion Coefficients D D kt = = µ µ q = therml voltge (5 T 300 o K) Einstein s Reltion Prof. Mrkus Zhn Pge 10 of 7

11 C. Drift-Diffusion Conuction Equilirium ( J = J = 0 ) J = 0 = ρ µ ED ρ = ρ µ Φ D ρ J = 0 = ρ µ E D ρ = ρ µ Φ D ρ D kt Φ = ρ = lnρ ρ µ q D kt Φ = ρ = lnρ ρ µ q ρ ρ q /kt = oe Φ ρ ρ q /kt = oe Φ Boltzmnn Distriutions ( =0 ) = ( =0 ) = o ρ Φ ρ Φ ρ [Potentil is zero when system is chrge neutrl] ( ρ ρ) ρo Φ Φ ρ ρ Φ q Φ q /kt q /kt o = = = e e = sinh kt (Poisson-Boltzmnn Eqution) Smll Potentil Approximtion: q Φ << 1 kt qφ sinh kt qφ kt ρ0q Φ Φ =0 kt Φ L Φ kt =0 ; L = ρ q o Deye Length Prof. Mrkus Zhn Pge 11 of 7

12 D. Cse Stuies 1. Plnr Sheet Φ Φ x/l =0 Φ =A1 e A e x L x/l B.C. Φ( x ± ) =0 ( x 0 ) = o Φ = ( x) Φ = x/l o e x > 0 < x/l o e x 0 Prof. Mrkus Zhn Pge 1 of 7

13 Φ E x = = x o e x/l x > 0 L o e x/l x < 0 L Ex ρ = = x e x 0 o x/l > L e x 0 o x/l < L σ ( = ) ( = ) s x=0 = Ex x 0 Ex x 0 = L o. Point Chrge (Deye Shieling) Φ 1 r r r Φ L 1 = r r r =0 Φ r ( Φ) E. Ohmic Conuction r Φ rφ r Φ =0 L r = A e A e r/l Φ 1 Q r = e 4π r r/l 0 r/l J = ρ µ ED ρ J = ρ µ E D ρ If chrge ensity grients smll, then ρ negligile ± ρ = ρ = ρ o J=J J = ρ µ ρ µ E= ρ µ µ E= σe o J= σ E (Ohm s Lw) σ = ohmic conuctivity Prof. Mrkus Zhn Pge 13 of 7

14 F. pn Junction Dioe Prof. Mrkus Zhn Pge 14 of 7

15 kt N Φ = Φ Φ = ln N A D n p q ni qn x qn x Φ ( x=0 ) = Φ p = Φn A p D n qnd x qna x n Φ = Φn Φp = p qn D xn qnd xn ND = ( xn xp) = 1 NA Prof. Mrkus Zhn Pge 15 of 7

16 II. Reltionship Between Resistnce n Cpcitnce In Uniform Mei Descrie y n σ. Di Ei qu S S C= = = v Eis Eis L Eis Eis v L L R= = = i Ji σ Ei S L S Eis Ei L L RC = = σ Ei Eis S L σ Check: Prllel Plte Electroes: l A R=, C= RC= σ A l σ Prof. Mrkus Zhn Pge 16 of 7

17 Coxil ln πl R=, C= RC= πσl ln σ Concentric Sphericl 1 1 R R 4 π 1 1 σ R R 1 R=, C= RC= 4 πσ 1 Prof. Mrkus Zhn Pge 17 of 7

18 III. Chnge Relxtion in Uniform Conuctors i ρ t u J u = 0 i E= ρ u J u = σ E ρu ρu σ σ i E = 0 ρu = 0 t t ρu τ = e σ = ielectric relxtion time ρ u t t τ ρu =0 ρu = ρ0 r, t=0 e τ e e IX. Demonstrtion Relxtion of Chrge on Prticle in Ohmic Conuctor Courtesy of Hermnn A. Hus n Jmes R. Melcher. Use with permission. Courtesy of Hermnn A. Hus n Jmes R. Melcher. Use with permission. Prof. Mrkus Zhn Pge 18 of 7

19 i i σ q q u J = σ E = = S S t q q =0 q=q t=0 e e τ = t τ e t τ ( e ) σ Prtilly Uniformly Chrge Sphere Courtesy of Krieger Pulishing. Use with permission. Prof. Mrkus Zhn Pge 19 of 7

20 ρ r < R ρ ( t=0 ) = Q = πr ρ 3 0 r > R 3 u T t τ ( t ) = e e r R = ρ ρ < τ σ u 0 1 e 0 r > R 1 r E r,t = t τe t τe ρ0 re Qre = 0 < r < R 3 4 π R Qe 4 π r t τe 0 Q 4 π r 3 1 R < r < R 1 r > R 1 σ ( r = R ) = E ( r = R ) E ( r = R ) su 0 r r t τe ( ) Q = 1 e 4 π R X. Self-Excite Wter Dynmos A. DC High oltge Genertion (Self-Excite) Prof. Mrkus Zhn Pge 0 of 7

21 Prof. Mrkus Zhn Pge 1 of 7

22 v st nc i v 1 = C v 1 = e nci = Cs t 1 1 v1 st nc i v = C v = e nci = Cs t 1 nci 1 1 Cs =0 nc i 1 Cs Det = 0 nci Cs nc C i ± =1 s= root lows up nci t C e Any perturtion grows exponentilly with time B. AC High oltge Self Excite Genertion v nc i v 1 = C ; v 1 = 1 e t v3 nc i v = C v = e t v1 nc i v 3 = C v 3 = 3 e t st st st nc i Cs nci Cs =0 Cs 0 nc i 3 et = 0 Prof. Mrkus Zhn Pge of 7

23 ( nc ) ( Cs ) =0 s= ( 1) 1 i nc C 3 3 i 3 s = nc C (exponentilly ecying solution) 1 i 13 1± 3j 1 = 1, ( ) nc C i s,3 = 1± 3 j (lows up exponentilly ecuse s rel >0 ; ut lso oscilltes t frequency s img 0) XI. Conservtion of Chrge Bounry Conition i ρ t u J u =0 i Ju ρu =0 S t ni J J σsu =0 t Prof. Mrkus Zhn Pge 3 of 7

24 XII. Mxwell s Cpcitor A. Generl Equtions E= _ E t i 0 < x < _ x E t i < x < 0 x E = v t =E t E t x x σsu ni J J =0 E ( t) E ( t) E ( t) E ( t ) σ σ =0 t t v t E ( t ) = E t v t v t σe ( t) σ E ( t) E ( t) E ( t) =0 t E σ σ v t v σ E ( t ) = t t Prof. Mrkus Zhn Pge 4 of 7

25 B. Step oltge: v ( t ) = u( t ) v = δ (n impulse) t Then ( t ) At t=0 E v = = t t t δ Integrte from t=0 - to t=0 0 E t t= 0 t= 0 δ t= 0 t t= 0 t= 0 ( = ) E t 0 = 0 t = E = t = E ( t = 0 ) = E ( t = 0 ) = For t > 0, v =0 t E σ σ σ E ( t ) = t σ t τ E ( t ) = A e ; τ = σ σ σ σ σ σ E ( t = 0 ) = A = A = σ σ σ σ σ t E ( t ) = ( 1 e ) e σ σ E t = E t τ t τ Prof. Mrkus Zhn Pge 5 of 7

26 σ ( t ) = E ( t) E ( t ) = E ( t) E ( t) su =E t ( σ σ ) ( σ σ ) = 1 e τ ( ) t C. Sinusoil Stey Stte: jω jω E t = Re E e jω E t = Re E e t t v t = Re e Conservtion of Chrge Interfcil Bounry Conition σ E ( t) σ E ( t) E ( t) E ( t ) = 0 t t E σ ω j E σ j ω = 0 E E = E E = E E σ jω σ j ω = 0 σ ω ( σ ω E ) σ ω j j = j = 0 E E = = jω σ jω σ ( σ ω ) ( σ ω ) j j σ = E E su ( ) σ σ ( σ ω ) ( σ ω ) = j j Prof. Mrkus Zhn Pge 6 of 7

27 D. Equivlent Circuit (Electroe Are A) I= j E A= j E A ( σ ω ) ( σ ω ) = R R R C jω 1 R C jω 1 R =, R = σ A σ A C =, C = A A Prof. Mrkus Zhn Pge 7 of 7

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