Download Conducting sphere inside electric field

Conducting sphere inside electric field
Submitted by: I.D. 310634175
The problem:
~ = E0 ẑ
A neutral conducting sphere with a radius R is placed inside a constant field E
1. Find the dipole induced by the sphere.
2. What is the electric field outside the sphere?
3. What is the charge distribution on the sphere shell?
The solution:
1. As we know the electric field inside a conducting sphere is 0.
The electric field induced inside a polarized sphere is
~ d = −k p~
E
R3
Therefore:
(1)
~z − E
~d = 0
E
k~
p
E0 ẑ − 3 = 0
R
(2)
(3)
(4)
and
R 3 E0
ẑ
(5)
k
The electric field outside the sphere is just the sum of the field induced by the sphere and the
constant external field:
p · ~r)~r − r2 p~
~ = k 3(~
E
+ E0 ẑ
(6)
r5
3pz z~r − r2 pz ẑ
r − r2 ẑ
3 3z~
= k
+
E
ẑ
=
E
R
+ ẑ
(7)
0
0
r5
r5
p~ =
Outside of the sphere the normal component of the electric field to the surface is
r − r2 ẑ
3 3z~
3 3z − r cos θ
~
Er = E · r̂ = E0 R
+ ẑ r̂ = E0 R
+ cos θ
r5
r4
2 cos θ
= E0 R 3
+ cos θ
r3
(8)
(9)
where θ is the angle between ~r and ẑ. Then at r = R
Er+ = 3E0 cos θ
(10)
whereas inside the sphere the field is zero.
By the Gauss’ law
~ r+ − E
~ r− = 4πkσ
E
(11)
Therefore,
σ=
3E0 cos θ
4πk
(12)
1