Chapter 4

Using Gauss’s law, we showed in Example 4.4 that the electric field due to the charge
distribution is
⎧ Q
⎪ rˆ,
r > a
2
⎪ 4πε0r
E = ⎨
⎪ Qr
rˆ,
r < a
3 ⎪⎩ 4πε0a
Figure 4.8.6
The electric potential at P1
(indicated in Figure 4.8.6) outside the sphere is
r Q 1 Q
V () r −V( ∞ ) =− ∫ dr′ = = k
r′ r
1
∞
2
4πε0 4πε0
On the other hand, the electric potential at P2
inside the sphere is given by
( ) ( )
2
∞ a ∞
2 3
4πε a
0r 4πε
0a
2 ⎛ ⎞
e
2 ⎛ ⎞
(4.8.3)
Q
r (4.8.4)
a r a Q r Qr
V () r −V( ∞ ) =− drE r > a − E r < a =− dr − dr′ r′
∫ ∫ ∫ ∫
1 Q 1 Q 1 2 2 1 Q r
= − 3 ( r − a ) = ⎜3− 2 ⎟
4πε 0 a 4πε 0 a 2 8πε
0 a ⎝ a ⎠
Q r
= ke
⎜3− 2 ⎟
2a⎝
a ⎠
A plot of electric potential as a function of r is given in Figure 4.8.7:
Figure 4.8.7 Electric potential due to a uniformly charged sphere as a function of r.
(4.8.5)
35