PHY 5346
HW Set 2 Solutions – Kimel
5. 2.2 The system is described by
a) Using the method of images
φx⃗ =
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 − 2xycosγ1/2
+
q ′
x2 + y ′2 − 2xy ′ cosγ1/2
σ = −q 1
4π
a 1 − y
2
a2
y2 + a2 − 2aycosγ3/2
Note
q induced = a2 ∫ σdΩ = −q 14π a
22πa 1 − y
2
a2
∫
−1
1
dx
y2 + a2 − 2ayx3/2
, where x = cosγ
q induced = −
q
2
aa2 − y2
2
aa2 − y2
= −q
c)
|F| =
qq ′
4π 0y ′ − y2
=
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
φx⃗ =
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.