Fields and Potentials of a Charged Conducting Sphere.

R
Q
Solid conducting sphere with
charge Q on the surface
(a) Find the electric field both inside and outside the sphere.
(b) Find the potential both inside and outside with the zero of potential at
infinity.
(a) Calculation of the fields:
E = 0
Inside: the field inside a conductor
must be zero under static conditions
Q
r
E
n
Outside: the field outside is found
using Gauss’ Law:
S
∫
S
Q
Q
E cos0
dA = ⇒
E
dA =
ε
∫
⇒
E =
0
ε
0
4
S
Q
r
πε
0
2
1

(b) Potential for r > R
s
r
s
We set the potential at infinity to be zero:
V
V −
V = −
r
r
0
∞
r
f
r
∫
∞
r r
E ⋅
ds = −
r
∫
∞
Qsˆ
4πε
s
0
ds
E
r
Qsˆ
E =
4πε
s
2
⋅
d ssˆ
Q
d s
Q ⎡1⎤
Q
= −
2
4πε
∫ = =
s 4πε
⎢
s ⎥
⎣ ⎦ πε
r
0 0 ∞
4
∞
r
0
0
2
No negative sign here
i
infinity
Potential inside the conducting sphere:
The potential at the surface of the conducting sphere
is found using the potential obtained for r > R by
setting r = R:
Q
V R
=
4πε
0
R
The potential inside the sphere is constant and must equal the potential
on its surface. Therefore the above result is also the potential
anywhere inside the sphere.
2