Introductory Physics Volume Two

2.1 Electric Potential 37
2 Electric Potential
§ 2.1 Electric Potential
The Coulomb force is a conservative force and thus we can consider
the potential energy of a particle in an electric field that is created by
a static charge distribution. Suppose that you have a charged particle
in an electric field, and that you move the particle from point A to
point B. As the particle is moved, the electric field will do work on the
particle.
W A→B =
=
= q
∫ B
A
∫ B
A
∫ B
A
⃗F E · ⃗dr
q ⃗ E · ⃗dr
⃗E · ⃗dr
Thus when the particle is moved from A to B the change in the electric
potential energy is
∫ B
∆U = −W A→B = −q ⃗E · ⃗dr
A
and the change in potential energy per charge is
∫ B
∆U
= − ⃗E ·
q
⃗dr.
A
Notice that the potential energy per charge does not depend on the
test charge q, it only depends on the electric field.
Definition: Electric Potential
The electric potential, V , is the electric potential energy per charge.
The electric potential is related to the electric field:
∫ B
V B − V A = ∆V = ∆U = −
q
A
For a uniform electric field this simplifies to
∆V = ∆U = −E q
⃗ · ∆⃗r
⃗E · ⃗dr