Jackson E&m Solution Homework 2, #2 .

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 q′ + y′ | y| |x⃗ − ⃗ |x⃗ − ⃗

with y ′ = ay , and q ′ = −q ay b) σ = − 0 ∂n∂ φ|x=a = + 0 ∂x∂ φ|x=a 2

σ = 0

1 ∂ 4π 0 ∂x

q q′ + 1/2 1/2 x 2 + y 2 − 2xy cos γ x 2 + y ′2 − 2xy ′ cos γ a 1−

y2 a2

σ = −q 1 4π y 2 + a 2 − 2ay cos γ 3/2 Note y2 q induced = a 2 ∫ σdΩ = −q 1 a 2 2πa 1 − 2 4π a

∫ −1 1

dx , where x = cos γ 3/2 y 2 + a 2 − 2ayx

q 2 q induced = − aa 2 − y 2  = −q 2 aa 2 − y 2  c) qq ′ q 2 ay 1 = , the force is attractive, to the right. 4π 0 a 2 − y 2  4π 0 y ′ − y 2 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 q′ q φx⃗ = 1 + +V 4π 0 |x⃗ − ⃗ y| y′ | |x⃗ − ⃗ and obviously the electric field in the sphere and induced charge on the inside of the sphere would remain unchanged. |F| =