Classical Electrodynamics - Jackson Problem 2.10.pdf

PHY 5346 HW Set 4 Solutions – Kimel 3. 2.10 As done in class we simulate the electric field E0 by two charges at ∞ σ = 30E0 cosθ a) This charge distribution simulates the given system for cosθ > 0.We have treated this probem in class. The potential is given by φx⃗ = −E0 1 − a 3 r3 r cosθ Using σ = −0 ∂∂n φ|surface We have the charge density on the plate to be σplate = −0 ∂∂z φ|z=0 = 0E0 1 − a 3 ρ3 For purposes of plotting, consider σplate 0E0 = 1 − 1 x3 0 0.2 0.4 0.6 0.8 1 2 4 6 8 10 x σboss = 30E0 cosθ = For plotting, we use σboss 30E0 = cosθ 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 b) q = 30E02πa2 ∫ 0 1 xdx = 3π 0E0a2 c) Now we have φr⃗ = 1 4π 0 q r⃗ − d⃗ + q ′ r⃗ − d⃗ ′ − q r⃗ + d⃗ − q ′ r⃗ + d⃗ ′ where q ′ = −q a d , d ′ = a 2 d . σ = −0 ∂∂r φ|r=a = −q 4π d2 − a2  a a⃗ − d⃗ 3 − d2 − a2  a a⃗ + d⃗ 3 q ind = 2πa2 −q 4π ∫0 1 d2 − a2  a a⃗ − d⃗ 3 − d2 − a2  a a⃗ + d⃗ 3 dx q ind = −qa2d2 − a2  2a 1 da 1 d − a − 1 a2 + d2 + 1 d + a − 1 a2 + d2 q ind = −12 q d2 − a2 d 2d d2 − a2 − 2 a2 + d2 = −q 1 − d2 − a2  d a2 + d2