Introductory Physics Volume Two

52 Electric Potential 2.7
And,
The potential is:
⃗E d = î(2E + ) + ĵ(0)
√ √
q 2 2q
E d = 2
2πɛ 0 a 2 2 = 2πɛ 0 a 2
V d = V + + V − = 1
4πɛ 0
(
q
( √ 2
2
)
(−q)
+
a) ( √ 2
2 a)
= 0
The potential is zero at both points! The electric field is not zero, and
it points in the x direction at both points. You should be able to argue
for any point lying on the line defined by the points c and d that the
electric field always points in the x direction and the potential is always
zero.
⊙ Do This Now 2.1
For the configuration in the previous example, compute the electric potential
at a point that is distance a above the +q charge.
q
8πɛ0 a(2 −
√
2)
Example
A charge Q is distributed uniformly along a line L that lies along the
x-axis. Compute the electric potential a distance x from the left end
of the charge distribution that is on the x-axis. Take the potential at
∞ to be zero.
Locate the origin at the point we want to compute the electric potential
(labeled P ):
x
P
x'
dx'
Consider the very small section of length dx ′ of the line that is at x ′ .
The amount of charge contained in this small section is
dq ′ = Q L dx′ .
Since the section is very small, we can treat it as a point charge and
the potential it causes at P is
dV ′ = 1 Q
L dx′
4πɛ 0 x ′ ,
where we have used the potential for a point charge, which takes the
potential at ∞ to be zero. To find the total electric potential at P we