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1 MIT OpenCourseWre /ESD.013J Electromgnetics nd Applictions, Fll 005 Plese use the following cittion formt: Mrkus Zhn, Erich Ippen, nd Dvid Stelin, 6.013/ESD.013J Electromgnetics nd Applictions, Fll 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 6.013, Electromgnetics nd Applictions Prof. Mrkus Zhn Septemer 7 nd 9, 005 Lectures 6 nd 7: Polriztion, Conduction, nd Mgnetiztion I. Experimentl Oservtion: Dielectric Medi A. Fixed Voltge - Switch Closed (v = 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 Vector P = N p = Nqd (p = q d dipole moment) N dipoles/volume (P is dipole density) d + q q Prof. Mrkus Zhn Pge 1 of 40

3 Courtesy of Krieger Pulishing. Used with permission. Prof. Mrkus Zhn Pge of 40

4 Q = inside V qnd i d = ρ dv p S V pired chrge or equivlently polriztion chrge density P S V V Q inside V = P i d = i P dv = ρ dv (Divergence Theorem) P= qnd = Np 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 40

5 i D = ρ u D i d = ρ u dv n i D D = σ su i P = ρ S V P P i d = ρ P dv n i P P = σ S V sp i (ε o E) = ρ u + ρ P ε o E i d = (ρ u + ρ S V P ) dv n i ε o E E = σ su + σ sp D. Polriztion Current Density Q = qn dv = 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 + ε E o t t = Ju + (ε oe + P) t D = Ju + ; D = ε 0 E + P t Prof. Mrkus Zhn Pge 4 of 40

6 III. Equipotentil Sphere in Uniform Electric Field lim Φ ( r, θ ) = E o r cos θ Φ= E o z = E o 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 r dipole o 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 40

7 IV. 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 Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission. For sphericl rry of non-intercting spheres (s >> R) 3 3 P=4 π ε R E _ o o i z P z =N p z =4 π εo R Eo N N= 1 s 3 R R P= ε 4 π 3 E= ψ ε 3 o e 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 3 ε = ε ε = ε 1 +ψ = ε 1 + π r o o e o 4 s Prof. Mrkus Zhn Pge 6 of 40

8 V. Demonstrtion: Artificil Dielectric Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission. Prof. Mrkus Zhn Pge 7 of 40

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 V = R s (ε ε ) R 3 o A A C = =4 π ε o d s d R=1.87 cm, s=8 cm, A= (0.4) m, d=0.15m ω =π(50 Hz), R s =100 k Ω, V=566 volts pek C=1.5 pf Prof. Mrkus Zhn Pge 8 of 40

10 v 0 = ω CR s V =(π) (50) (1.5 x 10-1 ) (10 5 ) 566 = volts VI. Plsm Conduction Model (Clssicl) dv + p m = q E m ν v n + dv p m = q E m ν v p = n kt, p = n kt + + k=1.38 x 10-3 joules/ o K Boltzmnn Constnt n 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 q E dj = d (q n v ) = q n dv = q n ( ) = q n E m m ω p ε ω = p + q+ n + q n, ω p = m + ε ( 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 Prof. Mrkus Zhn Pge 9 of 40

11 ω f = p 10 9x10 Hz p π B. Drift-Diffusion Conduction [Neglect inerti] 0 dv + (n+ k T ) q + k T m + = q + E m + ν + v + v + = E n + n+ m + ν + m + ν + n + 0 dv = q E n k T m m ν v ( ) v = q E k T n m ν m ν n n q+ n + q k T J + = q n v = E + n+ m + ν + m + ν + q n v = q n J = m ν E + q k T n m ν ρ + =q + n +, ρ = q n J = ρ µ E ρ D + + J = ρ µ E D ρ µ + = q +, D + = kt m + ν + m + ν + µ = q m ν, D = kt m ν chrge molulities Moleculr Diffusion Coefficients D + D = kt = = therml voltge (5 mv@ T 300 o K) µ + µ q Einstein s Reltion Prof. Mrkus Zhn Pge 10 of 40

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 40

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) = V o Φ ( x ) = d V e x/l o x > 0 V e +x/l d x < 0 o Prof. Mrkus Zhn Pge 1 of 40

14 V o e x/l d x > 0 dφ E x = = dx L d V o e x/l d x < 0 L d ε V o e x/l d x > 0 L d x ρ = ε de = dx ε V o e +x/l d x < 0 L d σs (x=0) = ε E x (x = 0 + ) E x (x = 0 ) = ε V o L d. Point Chrge (Deye Shielding) Φ =0 Φ L d d r Φ ( r Φ) =0 dr r /Ld 1 Φ r Φ = A1e + A e r r r r Φ ( r ) = Q e r/l d 4 π ε r = 1 (r Φ) r r L d 0 +r /L d E. Ohmic Conduction 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 40

15 F. p n Junction Diode Trnsition region P Chrge density ρ Electric field ε Potentil V V 0 An open-circuited pn diode kt NA N Φ = Φ n Φ p = ln q Figure y MIT OpenCourseWre. n i D 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 (x + x ε n p ) Prof. Mrkus Zhn Pge 14 of 40

16 Prof. Mrkus Zhn Pge 15 of 40

17 VII. 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 E i ds = ε σ Check: Prllel Plte Electrodes: l R=, σ A ε A C = l RC = ε σ Prof. Mrkus Zhn Pge 16 of 40

18 Coxil ln πεl R= πσ l, C= ln RC = ε 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 40

19 VIII. Chrge 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 = ρ (r, t=0 ) e t τe u 0 t τ e 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 40

20 σ q i σ u J d= E i d= ε S 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 40

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 Voltge Genertion (Self-Excited) 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: Krieger Pulishing Compny, ISBN: Prof. Mrkus Zhn Pge 0 of 40

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: Krieger Pulishing Compny, ISBN: Prof. Mrkus Zhn Pge 1 of 40

23 nc v = C dv v = V 1 e st i 1 1 i 1 nc v = C dv 1 v = V e st i i nc V = CsV nc V = CsV 1 nc i 1 V 1 Cs =0 nc i 1 Cs V 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 Voltge Self Excited Genertion From Electromgnetic Field Theory: A Prolem Solving Approch, y Mrkus Zhn, Used with permission. dv st nci v 1 = C ; v 1 = V 1 e nc v = C dv 3 v = V e st i nc v = C dv 1 v = V e st i nc i Cs 0 V 1 0 nc Cs V i = 0 Cs 0 nc i V 3 det = 0 Prof. Mrkus Zhn Pge of 40

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 dv S V = 0 d n i J J + σ su = 0 Prof. Mrkus Zhn Pge 3 of 40

25 XII. Mxwell Cpcitor A. Generl Equtions E = _ E t i 0 < x < ( ) x _ E t i < x < 0 ( ) x x x ( ) ( ) ( ) E d = v t =E t + E t J + dσ su n i J = 0 σ ( ) E ( ) v t E ( t ) = t σ E ( t ) σ E ( t ) σ E ( t ) + d ε E t ε E ( ) = 0 ( ) t ( ) ( ) + d ( ) ε E ( t ) ε v t E ( t) = 0 v t E t ε de σ + E t v t dv ε + σ + ( )= σ ( ) + ε Prof. Mrkus Zhn Pge 4 of 40

26 B. Step Voltge: v ( t ) = V u( t) Then dv =V δ ( t ) (n impulse) At t=0 dv = ε ε + ε de = ε Vδ ( t ) Integrte from t=0 - to t=0 + ε de ε + = ε + ε E = t= 0 + t 0 t= ε ε = Vδ ( t ) = t= 0 t= 0 V E ( t = 0 ) = 0 ε = ε ε V ε + E ( t = 0 + ) V E ( t = 0 + ) = ε + ε For t > 0, dv =0 ε de ε + σ + σ + E ( t ) = σ V σ ( ) V t τ E t = + A e σ + σ ε + ε ; τ = σ + σ ( ) = E t = 0 σ V ε + A = V ε A = V σ σ + σ ε + ε ε + ε σ + σ σ ( ) = V E t 1 σ + σ V E ( t ) = E ( t) ( e t τ ) + ε V ε + ε e t τ Prof. Mrkus Zhn Pge 5 of 40

27 σ ( ) ε E ( ) ε E ( t ) = ε E ( ) su t = t t ε V ε V =E ( t ) ε + ε V (σ ε σ = 1 e (σ + σ E ( ) t ε ) t τ ) ) ( C. Sinusoidl Stedy Stte: v ( t) = Re V e jω t E t = Re E e ( ) jωt ( ) e jωt E E t = Re 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 = V V E E = E V E = σ + j ω ε σ + ω ε = 0 j σ + ω ε + σ + j ω ε = V E σ + ω = j j ε = 0 ( ) E E V = = j ω ε + σ j ω ε + σ (σ + j ω ε ) + (σ + j ω ε ) σ su = ε E ε E = (εσ εσ ) V (σ + j ω ε ) + (σ + j ω ε ) Prof. Mrkus Zhn Pge 6 of 40

28 D. Equivlent Circuit (Electrode Are A) I = σ + j ω ε E A= σ ( ) ( ) + j ω ε E A = R V R + R C j ω+1 R C j ω+1 R =, R = σ A σ A ε A ε A C =, C = Courtesy of Krieger Pulishing. Used with permission. Prof. Mrkus Zhn Pge 7 of 40

29 XIII. Mgnetic Dipoles Courtesy of Krieger Pulishing. Used with permission. Courtesy of Krieger Pulishing. Used with permission. Prof. Mrkus Zhn Pge 8 of 40

30 Dimgnetism e I= π ω e ω e ω =, m= I π R i z = π π _ ω _ e R π R iz = i z i r i φ =m ωr e e ωr e i z e (r p) = m m e liner momentum Angulr Momentum L = m R i r v=m R ( ) L is quntized in units of h 34, h=6.6x10 π joule sec (Plnck s constnt) e L eh eh 4 m = = 9.3 x10 mp m m π m 4 π m e ( ) e = e Bohr mgneton m B (smllest unit of mgnetic moment) Imgine ll Bohr mgnetons in sphere of rdius R ligned. Net mgnetic moment is 4 A m = m B π R ρ 3 M 3 0 Totl mss of sphere 0 Avogdro s numer = 6.03 x 10 6 molecules per kilogrm mole moleculr weight For iron: ρ =7.86 x 10 3 kg/m 3, M 0 =56 Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission. Prof. Mrkus Zhn Pge 9 of 40

31 For current loop 4 3 A 0 4 A m=i π R =m 0 B πr ρ i = mb R ρ 3 M 0 3 M ( ) 3 i= For R = 10 cm ( ) 10 = 1.05 x 10 5 Amperes Thus, n ordinry piece of iron cn hve the sme mgnetic moment s current loop of rdius 10 cm of 10 5 Amperes current. B. Mgnetic Dipole Field H = µ _ 0 m 3 cos θ i r + _ sin θ i θ (multiply top & ottom y µ ) 0 4π r µ 0 Electric Dipole Field E = p i r 4 π ε 0 r 3 cos θ + sin θ i θ Anlogy p µ 0 m P= Np M=Nm, N = # of mgnetic dipoles / volume Polriztion Mgnetiztion Prof. Mrkus Zhn Pge 30 of 40

32 XIV. Mxwell s EQS Equtions with Mgnetiztion A. Anlogy to Mxwell s EQS Equtions with Polriztion EQS MQS i ( ε 0 E ) = ρ u i µ 0 H = i µ 0 M ρ p = i P (Polriztion or pired i P ( ) ( ) chrge density) n i ε 0 ( Eα E ) = n i P P + σ σ sp ρ = i (µ M) (mgnetic chrge m 0 density) ( ) ( ) 0 σ sm = n i µ 0 M M = n i P P ( ) H = J α su n i µ 0 H H = n i µ M M E = t µ 0 (H + M) Prof. Mrkus Zhn Pge 31 of 40

33 B. MQS Equtions B = µ 0 (H + M) Mgnetic flux density B hs units of Tesls (1 Tesl = 10,000 Guss) i B = 0 n i B B = 0 B E = t H = J v= dλ, λ = B i d (totl flux) S XV. Mgnetic Field Intensity long Axis of Uniformly Mgnetized Cylinder From Electromgnetic Fields nd Energy y Hermnn A. Hus nd Jmes R. Melcher. Used with permission. σ sm = n i µ 0 ( M M ) σ sm (z = d ) = µ 0 M 0 σ sm (z = d ) = µ 0 M 0 xh= J =0 H = Ψ 6.013, Electromgnetic Fields, Forces, nd Motion Lecture 6 & 7 Prof. Mrkus Zhn Pge 3 of 40

34 i( µ H ) = µ Ψ = ρ = i( µ M) 0 0 m 0 ρ ( ) ρ m (r ' ) dv ' Ψ = m Ψ r = µ 0 V' 4 πµ 0 r r ' Ψ ( ) R σ (z = d ) πr'dr ' R σ (z = d ) πr 'dr ' sm sm z = + r'=0 4 πµ 0 r r ' r'=0 4 πµ 0 r r ' R R µ 0 Mo πr'dr ' µ 0 Mo πr 'dr' = 1 1 r'=0 r'=0 4 πµ 0 r ' d + z 4 πµ 0 r ' d + z + 1 M 0 d Ψ ( z )= r' + z 1 r'dr ' 1 r' + (z + ) 1 = r' + (z + ) R r' = 0 M 0 d d = R + z z = r' 0 R 1 d r' + z + d R + z d z + Courtesy of Krieger Pulishing. Used with permission , Electromgnetic Fields, Forces, nd Motion Lecture 6 & 7 Prof. Mrkus Zhn Pge 33 of 40

35 d d M z z + 0 d z > 1 1 R + z d R + z + d Ψ H z = = z d d M z 0 z + d < z < d d R + z R + z + d XVI. Toroidl Coil Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission , Electromgnetic Fields, Forces, nd Motion Lecture 6 & 7 Prof. Mrkus Zhn Pge 34 of 40

36 Ni Ni Hi dl =H φ π r = Ni H = φ πr πr C Φ B π w 4 λ = N Φ =N B π w 4 Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission. V = i R = R H π R φ H H N 1 dλ dv v = =i R + V v = V v + R C (V =Horizontl voltge to oscilloscope ) v 1 dλ If R >> dv v R C λ = R C V V = Verticl voltge to oscilloscope C ω v ( v ) 1 π w V v = N B R C 4 π w = N B , Electromgnetic Fields, Forces, nd Motion Lecture 6 & 7 Prof. Mrkus Zhn Pge 35 of 40

37 Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission. Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission. Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission , Electromgnetic Fields, Forces, nd Motion Lecture 6 & 7 Prof. Mrkus Zhn Pge 36 of 40

38 XVII. Mgnetic Circuits Courtesy of Krieger Pulishing. Used with permission. In iron core: lim B = µ H µ H=0 B finite Hi dl = Hs = Ni H = Ni s Φ = µ 0 H Dd = µ 0 Dd N i s B i d = 0 S µ Dd λ µ 0 Dd λ = N Φ = 0 N i L= = N s i s 6.013, Electromgnetic Fields, Forces, nd Motion Lecture 6 & 7 Prof. Mrkus Zhn Pge 37 of 40

39 XVIII. Reluctnce Ni s (length) R = = = Φ µ 0 Dd (permeility)( cross sec tionl re ) [Reluctnce, nlogous to resistnce] A. Reluctnces In Series Courtesy of Krieger Pulishing. Used with permission. s R 1 = s 1, R = µ 1 D 1 µ D Ni Φ = R1 + R H i dl =H 1 s 1 + H s =Ni C 6.013, Electromgnetic Fields, Forces, nd Motion Lecture 6 & 7 Prof. Mrkus Zhn Pge 38 of 40

40 Φ = µ H D = µ H D µ Ni µ 1 1 Ni H 1 = ; H = µ s + µ s µ s + µ s B. Reluctnces In Prllel Ni H dl =H s =H s =Ni H =H = s i 1 1 C Φ = (µ H + µ ( 1 ) = Ni R + R H ) D = R R Ni (P + P ) P 1 = ; P = R 1 R 1 P = R [Permences, nlogous to Conductnce] XIX. Trnsformers (Idel) Courtesy of Krieger Pulishing. Used with permission , Electromgnetic Fields, Forces, nd Motion Lecture 6 & 7 Prof. Mrkus Zhn Pge 39 of 40

41 A. Voltge/Current Reltionships N i N i l Φ = 1 1 ; R = R µ A Another wy: i C H dl =Hl =N 1i 1 N i N i N i H= 1 1 l µ A Φ = µ HA = (N i N i ) = N i N i 1 1 l R 1 1 λ =N Φ = µ A (N i N N i ) =L i Mi l λ =N Φ = µ A (N N i N i ) = Mi + L i l µ A L 1 =N1 L 0, L =N L 0, M=N1 N L 0, L 0 = = 1 l R M= L L 1 1 dλ di di di di v = 1 1 = L M =NL N 1 1 N dλ di 1 di di 1 di v = = +M L =N L +N N 0 1 v 1 = N 1 v N i 1 N lim H 0 N i = N i = 1 1 µ i N v i 1 1 =1 v i 1 Courtesy of Hermnn A. Hus nd Jmes R. Melcher. Used with permission , Electromgnetic Fields, Forces, nd Motion Lecture 6 & 7 Prof. Mrkus Zhn Pge 40 of 40