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1 MIT OpenCourseWre Electromgnetic Fields, Forces, nd Motion, Spring 005 Plese use the following cittion formt: Mrkus Zhn, Electromgnetic Fields, Forces, nd Motion, Spring 005. (Msschusetts Institute of Technology: MIT OpenCourseWre). (ccessed MM DD, YYYY). License: Cretive Commons Attriution-Noncommercil-Shre Alike. Note: Plese use the ctul e you ccessed this mteril in your cittion. For more informtion out citing these mterils or our Terms of Use, visit:

2 6.641, Electromgnetic Fields, Forces, nd Motion Prof. Mrkus Zhn Lecture 7: Polriztion nd Conduction I. Experimentl Oservtion A. Fixed oltge - Switch Closed (v = o ) As n insulting mteril enters free-spce cpcitor t constnt voltge more chrge flows onto the electrodes; i.e. s x increses, i increses. B. Fixed Chrge - Switch open (i=0) As n insulting mteril enters free spce cpcitor t constnt chrge, the voltge decreses; i.e. s x increses, v decreses. II. Dipole Model of Polriztion A. Polriztion ector P = N p = Nqd (p = q d dipole moment) N dipoles/olume (P is dipole density) d q q Prof. Mrkus Zhn Pge 1 of 7

3 Courtesy of Krieger Pulishing. Used with permission. Prof. Mrkus Zhn Pge of 7

4 Q = inside qnd i d = ρ d P S pired chrge or equivlently polriztion chrge density P S Q inside = P i d = i P d = ρ d (Divergence Theorem) P=qNd i P = ρ P B. Guss Lw i (ε E) = ρ = ρ ρ = ρ i P o totl u P u unpired chrge density; lso clled free chrge density i (ε E P ) = ρ o u D = ε o E P Displcement Flux Density i D = ρ u C. Boundry Conditions Prof. Mrkus Zhn Pge 3 of 7

5 i D = ρ u D i d = ρ u d n i D D = σ su S i P = ρ P P i d = ρ P d n i P P = σ i (ε E) = ρ ρ S ε E i d = (ρ ρ sp ) d n i ε E E = σ o u P o u P o su S σ sp D. Polriztion Current Density Q = qnd = qndi d = P i d [Amount of Chrge pssing through surfce re element d] di p = Q = P i d t t [Current pssing through surfce re element d] = Jp i d polriztion current density P Jp = t Ampere s lw: x H = Ju Jp ε o E t = Ju P E ε o t t = Ju (ε oe P) t D = Ju t Prof. Mrkus Zhn Pge 4 of 7

6 III. Equipotentil Sphere in Uniform Electric Field ( ) o o o lim Φ r, θ = E r cos θ Φ = E z = E r cos θ r Φ (r = R, θ) = 0 Φ (r, θ) = E o r R 3 cos θ r This solution is composed of the superposition of uniform electric field plus the field due to point electric dipole t the center of the sphere: Φ = p cos θ with p = 4 πε E R 4πε o r dipole o o 3 This dipole is due to the surfce chrge distriution on the sphere. σ (r = R, θ) = ε E (r = R, s o r o θ) = ε Φ = ε o E o 1 R 3 r r R r = r= R = 3ε o E o cos θ 3 cos θ Prof. Mrkus Zhn Pge 5 of 7

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

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

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

10 I. Plsm Conduction Model (Clssicl) dv p m = q E m ν v n m dv = q E m ν v p n p = n kt, p = n kt k=1.38 x 10-3 joules/ o K Boltzmnn Constnt A. London Model of Superconductivity [ T 0, ν ± 0 ] dv m = q E dv, m = q E J = q n v, J = q n v dj d dv q n = (q n v ) = q n = q n ( q E ) = m m ω ε p E dj d dv q n m m ω ε p = (q n v ) = q n = q n ( q E ) = E ω p = q n q n m ε, ω p = ( ω p = plsm frequency) m ε For electrons: q - =1.6 x Couloms, m - =9.1 x kg 1 n - =10 0 /m 3, ε = ε o x10 frds/m ω = p q n m ε x10 rd/s f p = ωp 10 9x10 Hz π Prof. Mrkus Zhn Pge 9 of 7

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

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

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

14 o e x/l d x > 0 dφ E x = = dx L d o L d e x/l d x < 0 ε o e x/l d x > 0 L d x ρ = ε de = dx ε o e x/l d x < 0 L d σs (x=0) = ε E x (x = 0 ) E x (x = 0 ) = ε o L d. Point Chrge (Deye Shielding) Φ =0 Φ L d d r Φ ( r Φ) = 0 dr r /Ld 1 r Φ = A e A e r r r Φ r 1 Φ ( r ) = Q r /L e d 4 π ε r = 1 (r Φ) r r E. Ohmic Conduction L d 0 r /L d J = ρ µ E ρ D J = ρ µ E ρ D If chrge density grdients smll, then ρ ± negligile ρ = ρ = ρ o J = J J = (ρ µ ρ µ ) E= ρ o (µ µ ) E = σe J = σ E (Ohm s Lw) σ = ohmic conductivity Prof. Mrkus Zhn Pge 13 of 7

15 F. pn Junction Diode Prof. Mrkus Zhn Pge 14 of 7

16 kt NA ND Φ = Φ n Φ p = ln q n i Φ (x = 0 )= Φ p qn x A p ε = Φ n qn x D n ε qn D x n qn A x p Φ = Φ n Φ p = ε ε qn D x n qn D x n N D = (x n x p ) = 1 ε ε N A Prof. Mrkus Zhn Pge 15 of 7

17 II. Reltionship Between Resistnce nd Cpcitnce In Uniform Medi Descried y ε nd σ. q u C= v = S L Di d Ei ds = ε E i d L S E i ds v R= = i L S Ei ds Ji d = σ L E i ds S E i d RC = σ L Ei ds S Ei d ε L L E i d Ei ds = ε σ Check: Prllel Plte Electrodes: l R=, σ A ε A C = l RC = ε σ Prof. Mrkus Zhn Pge 16 of 7

18 Coxil ln πεl R=, C= RC = ε πσl ln σ Concentric Sphericl 1 1 R 1 R 4 πε R=, C= RC = ε 4 πσ 1 1 σ R 1 R Prof. Mrkus Zhn Pge 17 of 7

19 III. Chnge Relxtion in Uniform Conductors i Ju ρu = 0 t i E = ρ u ε J u = σ E σ i E ρ u ρ u σ = 0 ρ u = 0 t t ε ρ u ε τ = ε σ = dielectric relxtion time e ρ u ρ u =0 ρ u = ρ 0 (r, t=0 ) e t τ e t τe IX. Demonstrtion Relxtion of Chrge on Prticle in Ohmic Conductor Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission. Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission. Prof. Mrkus Zhn Pge 18 of 7

20 J i d= S σ q u E i d= = ε σ S dq dq q t e =0 q=q ( t=0 ) e τ (τ e = ε σ ) τ e Prtilly Uniformly Chrged Sphere Courtesy of Krieger Pulishing. Used with permission. Prof. Mrkus Zhn Pge 19 of 7

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

From Electromgnetic Field Theory: A Prolem

22 From Electromgnetic Field Theory: A Prolem Solving Approch, y Mrkus Zhn, Used with permission. Courtesy of Herert Woodson nd Jmes Melcher. Used with permission. Woodson, Herert H., nd Jmes R. Melcher. Electromechnicl Dynmics, Prt : Fields, Forces, nd Motion. Mlr, FL: Kreiger Pulishing Compny, ISBN: Prof. Mrkus Zhn Pge 1 of 7

23 nc v = C dv v = e st nc = Cs i i 1 nc v = C dv 1 v = e st nc = Cs i i 1 nc i 1 Cs 1 =0 nc i 1 Cs Det = 0 nc i nc i =1 s = ± root lows up Cs C nc i t e C Any perturtion grows exponentilly with time B. AC High oltge Self Excited Genertion dv st nci v 1 = C ; v 1 = 1 e dv 3 st nci v = C v = e nc v = C dv 1 v = e st i nc i Cs nc Cs i = 0 Cs 0 nc i 3 det = 0 From Electromgnetic Field Theory: A Prolem Solving Approch, y Mrkus Zhn, Used with permission. Prof. Mrkus Zhn Pge of 7

24 (nc i ) 3 Cs ) 3 nc ( = 0 s = i ( 1 C ) 1 3 s = nc C (exponentilly decying solution) 1 i 13 1 ± 3j (1) = 1, s = nc i 1 ± 3 j,3 (lows up exponentilly ecuse s rel >0 ; ut lso C oscilltes t frequency s img 0) XI. Conservtion of Chrge Boundry Condition ρu i Ju = 0 t J d d u i ρ u d S = 0 d n i J J σ su = 0 Prof. Mrkus Zhn Pge 3 of 7

25 XII. Mxwell s Cpcitor A. Generl Equtions E = _ E t i 0 < x < ( ) x _ E t i x < 0 ( ) x < Ex d x = v( t ) =E ( t) ( ) E t J dσ su n i J = 0 σ ( ) E ( ) v t E ( t ) = t E ( t ) σ E ( t ) d ε E t ε E ( ) = 0 ( ) t v( t ) d v ( ) ε E t ε ( t ) E t E t ( ) E ( t) = 0 σ ( ) σ ε de σ E t v t dv ε σ ( )= σ ( ) ε Prof. Mrkus Zhn Pge 4 of 7

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

27 σ ( ) ε E ( ) ε E ( t ) = ε E ( ) su t = t t ε ε =E ( t ) ε ε (σ ε σ = 1 e (σ σ E ( ) t ε ) t τ ) ) ( C. Sinusoidl Stedy Stte: ( ) = ( ) E t = Re E e jωt v t Re e jω t E ( t) = Re E e jωt Conservtion of Chrge Interfcil Boundry Condition σ E ( t ) σ E ( t ) d ε E ( t ) ε E ( t) = 0 E σ j ω ε E σ j ω ε = 0 E E = E E = E E σ j ω ε σ ω ε = 0 j σ ω ε (σ j ω ε ) E j = σ ω ε = 0 j E E = = j ω ε σ j ω ε σ (σ j ω ε ) (σ j ω ε ) σ su = ε E ε E (ε σ ε σ ) = (σ j ω ε ) (σ j ω ε ) Prof. Mrkus Zhn Pge 6 of 7

28 D. Equivlent Circuit (Electrode Are A) I = σ j ω ε E A= σ ( ) ( ) j ω ε E A = R R R C j ω1 R C j ω1 R =, R = σ A σ A ε A ε A C =, C = Prof. Mrkus Zhn Pge 7 of 7

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

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