Electric Potential and Electric Field

Consider a charge which is slowly moved an infinitesimal distance
along the -axis. Suppose that the difference in electric potential
between the final and initial positions of the charge is .
By definition, the change
in the charge's electric potential energy
is given by

(84) |

(85) |

(86) |

We call the quantity the

According to Eq. (87), electric field strength has dimensions
of potential difference
over length. It follows that the units of electric field are volts
per meter (
.
Of course, these new units are entirely equivalent to
newtons per coulomb: *i.e.*,

(88) |

Consider the special case of a uniform -directed electric field
generated by two uniformly charged parallel planes normal to the -axis. It is
clear, from Eq. (87), that if is to be constant between the plates
then must vary *linearly* with in this region. In fact, it is
easily shown that

According to Eq. (87), the -component of the electric field is equal
to minus the gradient of the electric potential in the -direction.
Since there is nothing special about the -direction, analogous rules
must exist for the - and -components of the field.
These three rules can be combined to give

We have seen that electric fields are superposable. That is, the electric
field generated by a set of charges distributed in space is
simply the *vector sum* of the electric fields generated by each charge
taken separately. Well, if electric fields are superposable, it follows
from Eq. (90) that electric potentials must also be superposable. Thus,
the electric potential generated by a set of charges distributed in space
is just the *scalar sum* of the potentials generated by each charge taken in isolation. Clearly, it is far easier to determine the potential generated by a set
of charges than it is to determine the electric field, since we can
sum the potentials
generated by the individual charges algebraically, and do not have to worry about
their directions (since they have no directions).

Equation (90) looks rather forbidding. Fortunately, however, it is possible
to rewrite this equation in a more appealing form. Consider two neighboring
points and . Suppose that
is the vector displacement of point relative to point .
Let be the difference in electric potential
between these two points.
Suppose that we travel from to by first moving a distance
along the -axis, then moving along the -axis,
and finally moving along the -axis. The net increase
in the electric potential as we move from to
is simply the sum of the increases as we move along the -axis,
as we move along the -axis, and as we move along the -axis:

(91) |

(92) |

where is the angle subtended between the vector and the local electric field . Note that attains its most negative value when . In other words, the direction of the electric field at point corresponds to the direction in which the electric potential decreases most rapidly. A positive charge placed at point is accelerated in this direction. Likewise, a negative charge placed at is accelerated in the direction in which the potential increases most rapidly (

In Sect. 4.3, we found that the electric field immediately above the surface of
a conductor is directed perpendicular to that surface. Thus, it is clear that the
surface of a conductor must correspond to an equipotential surface. In fact, since there
is no electric field inside a conductor (and, hence, no gradient in the electric
potential), it follows that the whole conductor (*i.e.*, both the surface and the
interior) lies at the same electric potential.