Electric Potential
Potential Difference and Electric Potential
When a test charge
q
o
is placed in an electric field
E
created by some source charge distribution, the electric force acting on the test charge is
q
o
E
. The force
q
o
E
is conservative because the force between charges described by Coulomb’s law is conservative. When the test charge is moved in the field by some external agent, the work done by the field on the charge is equal to the negative of the work done by the external agent causing the displacement d
s
. For an infinitesimal displacement d
s
of a charge, the work done by the electric field on the charge is F. d
s
=
q
o
E
. d
s
. As this amount of work is done by the field, the potential energy of the charge–field system is changed by an amount d
U
=
q
o
E
. d
s
. For a finite displacement of the charge from point
A
to point
B
, the change in potential energy of the system
∆
U
=
U
B

U
A
is The integration is performed along the path that
q
o
follows as it moves from
A
to
B
. Because the force
q
o
E
is conservative, this line integral does not depend on the path taken from
A
to
B
. The potential energy per unit charge
U/q
o
is independent of the value of
q
o
and has a value at every point in an electric field.
This quantity U/q
o
is called the electric potential
(or simply the potential)
V
. Thus, the electric potential at any point in an electric field is
The potential difference
∆∆∆∆
V= V
B
 V
A
between two points
A
and
B
in an electric field is defined as the change in potential energy of the system when a test charge is moved between the points divided by the test charge
q
o
: Potential difference should not be confused with difference in potential energy. The potential difference between
A
and
B
depends only on the source charge distribution (consider points
A
and
B without
the presence of the test charge), while the difference in potential energy exists only if a test charge is moved between the points.
Electric potential is a scalar characteristic of an electric field, independent of any charges that may be placed in the field
. If an external agent moves a test charge from
A
to
B
without changing the kinetic energy of the test charge, the agent performs work which changes the potential energy of the system:
W =
∆∆∆∆
U
. The work done by an external agent in moving a charge
q
through an electric field at constant velocity is The
SI
unit of both electric potential and potential difference is
joules per coulomb
, which is defined as a
volt
(V): That is,
1 J
of work must be done to move a
1C
charge through a potential difference of
1 V
.
Potential Differences in a Uniform Electric Field
Consider a uniform electric field directed along the negative
y
axis, as shown in Figure 1a. Let us calculate the potential difference between two points
A
and
B
separated by a distance
s = d
, where
s
is parallel to the field lines. Because
E
is constant, we can remove it from the integral sign; this gives The negative sign indicates that the electric potential at point
B
is lower than at point
A
; that is,
V
B
<
V
A
.
Electric field lines always point in the direction of decreasing electric potential, as shown in Figure 1a.
Figure (1)
Now suppose that a test charge
q
o
moves from
A
to
B
. The change in the potential energy of the charge–field system is given by
From this result, we see that :
(1)
if
q
o
is positive, then
∆∆∆∆
U
is negative. We conclude that a system consisting of a positive charge and an electric field loses electric potential energy when the charge moves in the direction of the field. This means that an electric field does work on a positive charge when the charge moves in the direction of the electric field.
(2)
If
q
o
is negative, then
∆∆∆∆
U
is positive and the situation is reversed: A system consisting of a negative charge and an electric field gains electric potential energy when the charge moves in the direction of the field.
Figure (2)
Now consider the more general case of a charged particle that moves between
A
and
B
in a uniform electric field such that
the vector
s
is not parallel
to the field lines, as shown in Figure 2. In this case the potential difference is given by where again we are able to remove
E
from the integral because it is constant. The change in potential energy of the charge–field system is