6.641,Electromagnetic Fields,Forces,and Motion Prof.Markus Zahn Lecture 3:Electroquasistatic and Magnetoquasistatic Fields and Boundary Conditions I.Conditions for Electroquasistatic Fields A.Order of Magnitude Estimate [Characteristic Length L,Characteristic timet Figure 3.3.1 Prototype systems involving one typical length.(a)EQS system in which source of EMF drives a pair of perfectly conducting spheres having radius and spacing on the order of L. (a) Courtesy of Hermann A.Haus and James R.Melcher.Used with permission. .E=e是=peE=g xH=εE-H=E=H-eEL-p t VxE Eomg >==一Eo2L2 L Eipt 2 Femor=HpL3 2 (cr)2 i c=-1 VEμ Eeor《1《1 E 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 1 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 1 of 12 6.641, Electromagnetic Fields, Forces, and Motion Prof. Markus Zahn Lecture 3: Electroquasistatic and Magnetoquasistatic Fields and Boundary Conditions I. Conditions for Electroquasistatic Fields A. Order of Magnitude Estimate [Characteristic Length L, Characteristic time τ ] Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. E L E E L ρ ∇ =ρ ⇒ =ρ ⇒ = ε ε ε i 2 E H E EL L H H t L ∂ ρ ∇× = ⇒ = ⇒ = = ∂ τ ττ ε ε ε 2 3 error 2 error H HL L E E E t L ∂ µ µρ µρ ∇× = − ⇒ = = ⇒ = ∂τ τ τ µ ( ) 3 22 error 2 2 E L LL 1 ; c E L c µρ µ = == τρ τ τ µ ε ε = ε error E L 1 1 E c ⇒ τ
B.Estimate of Error introduced by EQS approximation Z=d Z-0 Figure 3.3.2 Plane parallel electrodes having no resistance,driven at their outer edges by a distribution of sources of EMF. Courtesy of Hermann A.Haus and James R.Melcher.Used with permission. E=0=6 -Eo z=d Osu= +Eo z=0 K2b+b2du=0→K,=-bdcu=-bεdE dt 2 dt 2 dt .s-民阔-H2r=ms胎-4= dEo dt dt fE.ds=-「u H.da Figure 3.3.3 Parallel plates of Figure 3.3.2,showing volume containing lower plate and radial surface current density at r=0 r=b its periphery. Courtesy of Hermann A.Haus and James R.Melcher.Used with permission. Z Z=d Figure 3.3.4 Cross-section of system z=0 shown in Figure 3.3.2 showing surface and contour used in evaluating correction E field. Courtesy of Hermann A.Haus and James R.Melcher.Used with permission. 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 2 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 2 of 12 B. Estimate of Error introduced by EQS approximation Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. _ _ z z 0 V E i Ei d = = 0 su 0 E zd E z0 ⎧⎪− = σ = ⎨ ⎪+ = ⎩ ε ε 2 su su 0 r r d dd b b E K2 b b 0 K dt 2 dt 2 dt σ σ π +π = ⇒ =− =− ε ( ) 2 0 0 C S dE r dE H ds E da H 2 r r H t dt 2 dt φ φ ∂ = ⇒π = π ⇒= ∂ ε εε ∫ ∫ i i v C S H E ds da t ∂ =− µ ∂ ∫ ∫ i i v Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. Courtesy of Hermann A. Haus and James R. Melcher. Used with permission
[店o-6]a=+竖jr'ar'a dt2 6-6+号眼-的) If Eo(t)=Acos@t Eml-2-r2）-422-r） Eo 4Eo dt2 Eerrortu -1 Eo 4 f=c=1 Veμ 2-60-2受心-爱Px1bx月 入 4 f=1 MHz in free space3x108 =300m 106 Ifb≤100m EQS approximation is valid. II.Conditions for Magnetoquasistatic Fields H (b)MQS system consisting of perfectly conducting loop driven by current source.The radius of the loop and diameter of its cross-section are (b) on the order of L. Courtesy of Hermann A.Haus and James R.Melcher.Used with permission. 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 3 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 3 of 12 ( ) () b 2 0 z z 2 r d E E b E r d r ' dr ' d 2 dt µ ⎡ ⎤ − =+ ⎣ ⎦ ε ∫ ( ) 2 2 2 0 2 d d E b r 4 dt µ = − ε ( ) ( ) 2 0 2 2 z 0 2 d E Er E r b 4 dt µ =+ − ε If E t A cos t 0 ( ) = ω ( ) ( ) 2 error 0 22 2 22 2 0 0 E d E 1 br br E 4E 4 dt µ = − =ω − ε εµ 2 2 error 0 E b 1 1 E 4 ω µ ⇒ ε 1 f c λ= = εµ 22 2 2 2 2c b c b 1b 2 4 ω π ωµ π λ λ= ⇒ω= ⇒ = ⇒ πλ π λ ε f=1 MHz in free space 8 6 3 10 300m 10 × ⇒λ= = If b 100m EQS approximation is valid. Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. II. Conditions for Magnetoquasistatic Fields
x府=j-H=JH=J儿 95_H=E=HL- vxm=6要-些Ho-县- t L t 2 2 Herror._8JL3_8uL2-L☑ M2之（c2C quasistatic Figure 3.4.1 Range of characteristic times over which quasistatic approximation is valid.The transit time of an Tem 入 electromagnetic wave is Tem while r?is a time characterizing the dynamics of the quasistatic system. Courtesy of Hermann A.Haus and James R.Melcher.Used with permission. L =LVEH Tem =C 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 4 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 4 of 12 H H J J H JL L ∇× = ⇒ = ⇒ = 2 H E H HL JL E E t L ∂ µ µµ ∇ × = −µ ⇒ = ⇒ = = ∂ τ ττ 3 error error error 2 2 E E EL JL H H H t L ∂ µ ∇× = ⇒ = ⇒ = = ∂ τ τ τ ε εε ε ( ) 3 22 error 222 H JL L L 1Lc H JL c µ µ = == ⇒τ τ τ τ ε ε Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. em L L c τ== µε
III.Boundary Conditions 1.Gauss'Continuity Condition 才 ds=nds ，+ 十 + n·co(E2-E)=o Y ds =-nds Figure 2-19 Gauss's law applied to a differential sized pill-box surface enclosing some surface charge shows that the normal component of eoE is discontinuous in the surface charge density. Courtesy of Krieger Publishing.Used with permission. Ed-[adS-E)dS-a.dS 8(En-E)=a→n…[8(2-】=a 2.Continuity of Tangential E E ←E2t nx(E2-E1)=0 (a) Figure 3-12 (a)Stokes'law applied to a line integral about an interface of dis- continuity shows that the tangential component of electric field is continuous across the boundary. Courtesy of Krieger Publishing.Used with permission. ∮E.ds=(Et-E2x)dl=0→Et-Ex=0 n×(尼-E2)=0 Equivalent toΦ，=Φ2 along boundary 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 5 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 5 of 12 III. Boundary Conditions 1. Gauss’ Continuity Condition Courtesy of Krieger Publishing. Used with permission. 0 s 0 2n 1n s ( ) S S v∫ ∫ ε ε E da = dS E - E dS = dS i σ σ ⇒ 0 2n 1n s 0 s ( ) E -E = n E -E = ( 2 1 ) ⇒ ⎡ ⎤ ⎣ ⎦ ε ε σ i σ 2. Continuity of Tangential E Courtesy of Krieger Publishing. Used with permission. ( ) 1t 2t 1t 2t C E ds = E - E dl = 0 E - E 0 ⇒ = ∫ i v n× E -E = 0 ( 1 2 ) Equivalent to Φ Φ 1 2 = along boundary
3.Normal H ∮，H.a=0 (a) Area A Ho (Han -Hpn)A =0 Han Hon n[。-i]=0 (b) Hb 4.Tangential H .ds=ja+品，E. Ha (a) Hptds-Hatds=Kds n Hpt Hat =K ds Surface current K out nx[i。-H。]=K of page (amp/meter) (b) 5.Conservation of Charge Boundary Condition (a) 个方 ∮j.a+paw=0 dt (b) n[。-j]+0,=0 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 6 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 6 of 12 3. Normal H 0 S µ H da = 0 ∫ i v µ0 an bn (H -H A 0 ) = H =H an bn n H - H = 0 a b ⎡ ⎤ ⎣ ⎦ i 4. Tangential H 0 CS S d H ds = J da E da dt + ε ∫∫ ∫ ii i v H ds - H ds = Kds bt at H -H =K bt at n × H - H = K a b ⎡ ⎤ ⎣ ⎦ 5. Conservation of Charge Boundary Condition S V d J da + dV = 0 dt ρ ∫ ∫ i v n J - J + = 0 a b s t ∂ ⎡ ⎤ ⎣ ⎦ ∂ i σ
6.Electric Field from a Sheet of Surface Charge a.Electric Field from a Line Charge 00 dq1 =Xodz (r2+z2)1/2 dE2 dEdE:+dE2 r2+z21/2de1 dq2=入odz 一00 An infinitely long uniform distribution of line charge only has a radially directed electric field because the z components of the electric field are canceled out by symmetrically located incremental charge elements as shown above. 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 7 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 7 of 12 6. Electric Field from a Sheet of Surface Charge a. Electric Field from a Line Charge
dq d-4o2+2) 0s0= Xordz 4o2+z2为 05 =422+2 0 三2 Another way:Gauss'Law 入0 个N Gaussian Surface [eoE.da=eoE,2πrl=oL Er= 10 2or 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 8 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 8 of 12 ( ) ( ) 0 r 2 2 3 2 2 2 0 0 dq rdz dE cos 4 rz 4 rz λ = θ = π + π + ε ε ( ) 0 r r 3 0 2 2 2 z z r dz E dE 4 r z +∞ +∞ = −∞ = −∞ λ = = π + ε ∫ ∫ ( ) 0 1 0 22 2 2 z r z 4 rz r +∞ =−∞ λ = π + ε 0 2 r0 λ = πε Another way: Gauss’ Law ε ε = π =λ ∫ i 0 0 r S E da E 2 rL L 0 λ = πε 0 r 0 E 2 r
b.Electric Field from a Sheet Charge d=cndx (x2+y2)1/2 dE=dE,+dE2 26 (a) (b) 个do=Pody Po 川 Poa 280 →y dE=- 2 0 (c) (a)The electric field from a uniformly surface charged sheet of infinite extent is found by summing the contributions from each incremental line charge element.Symmetrically placed line charge elements have x field components that cancel,but y field components that add.(b)Two parallel but oppositely sheets of surface charge have fields that add in the region between the sheets but cancel outside.(c)The electric field from a volume charge distribution is obtained by summing the contributions from each incremental surface charge element. d dEy = C0s0= coydx 2(x2+y2 2πeo(x2+y2] -I t dx x2+y2 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 9 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 9 of 12 ( ) ( ) 0 y 1 2 2 2 2 2 0 0 d ydx dE cos 2 xy 2 xy λ σ = θ = π + π + ε ε 0 y y 2 2 0 x x y dx E dE 2 x y +∞ +∞ = −∞ = −∞ σ = = πε + ∫ ∫ 0 1 0 y 1 x tan 2y y +∞ − −∞ σ = πε b. Electric Field from a Sheet Charge
o y>0 2E0 60 y-a ooiy 280 280 y>a oo iy 2 2E0 ya 6.641,Electromagnetic Fields,Forces,and Motion Lecture 3 Prof.Markus Zahn Page 10 of 12
6.641, Electromagnetic Fields, Forces, and Motion Lecture 3 Prof. Markus Zahn Page 10 of 12 0 0 0 0 y 0 2 y 0 2 ⎧ σ > ⎪ ⎪ = ⎨ σ ⎪− − − > = = ⎨ ⎨ ⎪ ⎪ σ σ −