(173) |

(174) | |||

Since there is nothing special about the -axis, we can write

where is a differential operator which involves the components of but not those of . It follows from Eq. (172) that

where

Thus, the electric field generated by a collection of fixed charges can be written as the gradient of a scalar potential, and this potential can be expressed as a simple volume integral involving the charge distribution.

The scalar potential generated by a charge located at the origin is

(178) |

(179) |

Thus, the scalar potential is just the sum of the potentials generated by each of the charges taken in isolation.

Suppose that a particle of
charge is taken along some path from point to point .
The net work done on the particle by electrical forces is

(181) |

Thus, the work done on the particle is simply minus its charge times the difference in electric potential between the end point and the beginning point. This quantity is clearly independent of the path taken between and . So, an electric field generated by stationary charges is an example of a conservative field. In fact, this result follows immediately from vector field theory once we are told, in Eq. (176), that the electric field is the gradient of a scalar potential. The work done on the particle when it is taken around a closed loop is zero, so

(183) |

for any electric field generated by stationary charges. Equation (184) also follows directly from Eq. (176), since for any scalar potential .

The SI unit of electric potential is the volt, which is equivalent to a joule per coulomb. Thus, according to Eq. (182), the electrical work done on a particle when it is taken between two points is the product of its charge and the voltage difference between the points.

We are familiar with the idea that a particle moving in
a gravitational field possesses potential energy as well as kinetic
energy. If the particle moves from point to a lower point then the
gravitational field does work on the particle causing its kinetic energy to
increase. The increase in kinetic energy of the particle is balanced by an
equal decrease in its potential energy, so that the overall energy of the
particle is a conserved quantity. Therefore, the work done on the particle
as it moves from to is *minus* the difference in its gravitational
potential energy between points and . Of course, it only makes sense to
talk about gravitational potential energy because the gravitational field
is conservative. Thus, the work done in taking a particle between two
points is path independent, and, therefore, well-defined. This means that the
difference in potential energy of the particle between the beginning and end
points is also
well-defined.
We have already seen that
an electric field generated by stationary charges is a conservative field.
In follows that
we can define an electrical potential energy of a particle moving in such a field.
By analogy with gravitational fields, the work done in taking a particle
from point to point is
equal to minus the difference in potential energy of the particle between
points
and . It follows from Eq. (182), that
the potential energy of the particle at a general
point , relative to some reference point (where the potential energy is set to zero), is given by

(185) |

The scalar electric potential is undefined to an additive
constant. So, the transformation

(186) |