Estimating Static Fields Examples of point and line Images of lightning rod, power line corona, and 0.1 micron field integrated circuits removed due to copyright restrictions. emission Φ=bEdz − ab ∫a i ∝ r1 (spherical case), or ∝ ln r (cylindrical case) E ||∝2 L EEr− ρο r ρο r ||Er∝−1 R R Surface area = 2πrLSurface = 4πr2 ∞ Spherical geometry Cylindrical geometry L7-1L7-2 2 2 2 2 4ˆ 4 ( ) ( ) = 4 4 • πε = = π ε ⇒ = = πε πε ⇑ ∫ a a r rA V a VaQQ Dnda r Er Er r r r 2 11( ) 4 4 4 4 ⎛ ⎞ = = = − − ≅ ⇒ = πε ⎜ ⎟πε πε πε⎝ ⎠∫ ∫b b a r aa a Q dr Q QV Erdr Q V a ba ar (b >> a) a b Simple Static Example Spherical breakdown Q V Lightning rod: a ≅ 1 mm, V 104 volts ⇒ 107 V/m breakdown/corona Power line: a ≅ 1 cm, V 105 volts ⇒ 107 V/m breakdown/corona Integrated Circuit: a ≅ 0.1μm, V = 1 volt ⇒ 107 V/m breakdown/corona > > ()= ⇑ a r VEa a () 4 = πε QVr r ⇓L7-3 SOLVING FOR STATIC FIELDS Approaches when given Φ, Ψ on boundaries: q q p V pq Use dv, E4r ρ Φ= = −∇Φ πε∫ q qp pq 2V pq ˆE r dv 4r ρ = πε∫ q p q p q2V p q J(r -r ) H dv 4| r -r | × = π∫ Approaches when given ρ, J: z y x dVq rpq Vq rq rp p Superposition integrals Biot-Savart Law Use Laplace’s Equation. Derivation: 2BElectric : ×E = - = 0 (statics) E = - Φ. ρ =0 E = 0 Φ =0 t ∂∇ ⇒ ∇ ⇒ ∇ ⇒ ∇ ∂ i . 2DMagnetic : ×H = = 0 (statics; J = 0) H = -H = 0 = 0 t ∂∇ ⇒ ∇Ψ ∇ ⇒ ∇ Ψ ∂ iSEPARATION OF VARIABLES Static charge-free regions obey Laplace’s equation: Electric potential ∇2Φ(r) = 0 ; Magnetic potential ∇2Ψ(r) = 0 Assume Φ(x,y) = X(x)Y(y) ⇒ 2 2 2 2 2 ∂Φ ∂Φ dX dY∇Φ= + =Y(y) + X(x) = 0 ∂x2 ∂y2 dx2 dy2 2 21dX =−1dY =−k2 "separation constant" Xdx 2 Y dy 2 Solutions: 2dX 2=−k X⇒ X(x) =Acos(kx) +Bsin(kx) dx2 For k2 > 0;2 (swap x,ydY 2 = +Dsinh(ky)=kY⇒ Y(y)Ccosh(ky)dy2 If k2 < 0) or: Y(y) = C'eky + D'e-ky (equivalent to above) L7-4SEPARATION OF VARIABLES Solution to Laplace’s equation when k2 = 0: Φ(x,y) = X(x)Y(y) ⇒ Φ(x,y) = (Ax + B)(Cy + D) Example, Cartesian coordinates: {Φ = (Ax+B)(Cy+D) = 0 at x=0 for all y} ⇒ B = 0 Given Φ(x,y) Φo Φ=0 Φ(x,y) 0 y xLx 2 2 0 Xdx Y dy = − = Ly 2 21d X 1 d Y 2 2 2 dx 2 dX 0= X(x) = Ax + B dY 0 Y(y) = Cy + D ⇒ dy ⇒ = {Φ = (Ax+B)(Cy+D) = 0 at y=0 for all x} ⇒ D = 0 {Φ = Φo at x=Lx, y=Ly} ⇒ AC = Φo/LxLy Φ = xyΦo/LxLy Φ is matched at all 4 boundaries L7-5CIRCULAR COORDINATES Separation of Laplace’s equation: Only in cartesian, cylindrical, spherical, and elliptical coordinates Circular coordinates: 2 r2 (r, ) 1 ∂ (r ∂Φ ) + 12 (∂Φ ) = 0 θ∇Φ θ = 2rr∂∂r r ∂θ rd dR 1d2Θ 2 x R(r) ( ) ⇒ (r ) =− ( 2) = mΦ= Θθ Rdr dr Θ dθ Solutions (pick the one matching boundary condition): (r, ) (A + θ + Dln r) for m2 = 0Φθ= B )(C (r, ) (Asin m θ+ θ m + Dr −m) for m2Φθ= Bcosm )(Cr > 0 Φθ= (r, ) [Asinh m θ + Dsin( 2θ+ Bcosh m )[Ccos(mln r) mln r)] for m < 0 Example – conducting cylinder (m = 0): V volts rΦ(r,θ) = C+D ln r = V[2-(lnr/lnR)](r ≥ R); Φ(r,θ) = V (r < R) 0 RE(r) =−∇Φ=− (r ˆ ∂ +θ ˆ1 ∂ )Φ= V [V/m] (r > R) ∂r r ∂θ rlnR σ = ∞ L7-6L7-7 ⎡ ⎤+ ε−ε ⎢ ⎥ρ = −∇ • = −∇ • ε − ε = − ε − ε = − ⎢ ⎥σ σ ⎢ ⎥⎣ ⎦ o 3o o p o o o o xJ(1 ) ( )J d LP ( )E ( ) [C/m ] dx L INHOMOGENEOUS MATERIALS Governing Equations: o f pJ E D E E P D P=σ =ε =ε + ∇• =ρ ∇• =−ρ Non-uniform Conductivity σ(x) (e.g., doping gradients in pn junctions): Assume o [S/m] x1 L σσ= + x 0 I [A] J [A/m2] L o o xJ(1 )J LˆE x [V/m] + = = σ σ ⎡ ⎤+ ε⎢ ⎥ρ=∇• =∇•ε =ε = ⎢ ⎥σ σ ⎢ ⎥⎣ ⎦ o 3o f o o xJ(1 ) Jd LD E [C/m ] dx L Note: Non-uniform conductors have free charge density ρf and polarization charge density ρp throughout.L7-8 INHOMOGENEOUS PERMITTIVITY Governing Equations: o f pJ E D E E P D P=σ =ε =ε + ∇• =ρ ∇• =−ρ Example, Non-uniform Permittivity ε(x): Assume: o x(1 ) [F/m] Lε=ε + Non-uniform dielectrics have polarization charge density ρp throughout. x 0 L +V 0 volts o o o o D D ED E f(x) E x x(1 ) (1 )L L =ε ≠ ⇒ = = = ε ε + + o x o o L L 0 0 E VV E dx dx E Lln2 [V] E = [V/m] x L ln2 1 L = = = ⇒ + ∫ ∫ VˆTherefore : E x [V/m] xLln2(1 )L = + − −ρ = −∇ • = −∇ • = ∇ = + = + 1 p o o 2 2 d x VP (D-ε E) ε •E (V/Lln2) (1 )dx L xLln2(1 )L Note: Non-uniform E(x) ε(x)MIT OpenCourseWare http://ocw.mit.edu 6.013 Electromagnetics and Applications Spring 2009 For information about citing these materials or our Terms of Use, visit:
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