Classical Electrodynamics - Jackson Problem 2.2.pdf

PHY 5346 HW Set 2 Solutions – Kimel 5. 2.2 The system is described by a) Using the method of images φx⃗ = 1 4π 0 q |x⃗ − y⃗| + q ′ |x⃗ − y⃗ ′ | with y ′ = a 2 y , and q ′ = −q ay b) σ = −0 ∂∂n φ|x=a = +0 ∂ ∂x φ|x=a σ = 0 1 4π 0 ∂ ∂x q x2 + y2 − 2xycosγ1/2 + q ′ x2 + y ′2 − 2xy ′ cosγ1/2 σ = −q 1 4π a 1 − y 2 a2 y2 + a2 − 2aycosγ3/2 Note q induced = a2 ∫ σdΩ = −q 14π a 22πa 1 − y 2 a2 ∫ −1 1 dx y2 + a2 − 2ayx3/2 , where x = cosγ q induced = − q 2 aa2 − y2  2 aa2 − y2  = −q c) |F| = qq ′ 4π 0y ′ − y2 = 1 4π 0 q2ay a2 − y2 , the force is attractive, to the right. d) If the conductor were fixed at a different potential, or equivalently if extra charge were put on the conductor, then the potential would be φx⃗ = 1 4π 0 q |x⃗ − y⃗| + q ′ |x⃗ − y⃗ ′ | + V and obviously the electric field in the sphere and induced charge on the inside of the sphere would remain unchanged.