We know that a static electric field is conservative, and can consequently
be written in terms of
a scalar potential:

(576) |

(577) |

The negative sign in the above expression comes about because we would have to exert a force on the charge, in order to counteract the force exerted by the electric field. Recall that the scalar potential generated by a point charge at position is

Let us build up our collection of charges one by one. It takes no work to bring the
first charge from infinity, since there is no electric field to fight against.
Let us clamp this charge in position at . In order to bring the
second charge into position at ,
we have to do work against the electric field
generated by the first charge. According to Eqs. (578) and Eqs. (579),
this work is given by

(581) |

(582) |

(583) |

This is the

Equation (584) can be written

is the scalar potential experienced by the th charge due to the other charges in the distribution.

Let us now consider the potential energy of a continuous charge distribution.
It is tempting to write

is the familiar scalar potential generated by a continuous charge distribution. Let us try this out. We know from Maxwell's equations that

(589) |

(590) |

(591) |

(592) |

where is some volume which encloses all of the charges, and is its bounding surface. Let us assume that is a sphere, centred on the origin, and let us take the limit in which the radius of this sphere goes to infinity. We know that, in general, the electric field at large distances from a bounded charge distribution looks like the field of a point charge, and, therefore, falls off like . Likewise, the potential falls off like . However, the surface area of the sphere increases like . Hence, it is clear that, in the limit as , the surface integral in Eq. (593) falls off like , and is consequently zero. Thus, Eq. (593) reduces to

where the integral is over all space. This is a very nice result. It tells us that the potential energy of a continuous charge distribution is stored in the electric field. Of course, we now have to assume that an electric field possesses an

We can easily check that Eq. (594) is correct. Suppose that we have a
charge which is uniformly distributed within a sphere of
radius . Let us imagine building up this charge distribution
from a succession of thin spherical layers of infinitesimal thickness. At each
stage, we gather a small amount of charge from infinity, and spread it
over the surface of the sphere in a thin
layer from to . We continue this process until the final radius of the
sphere is . If is the charge in the sphere when it has attained radius
, then the work done in bringing a charge to it is

(597) |

(598) |

(599) |

(600) |

Now that we have evaluated the potential energy of a spherical charge distribution
by the direct method, let us work it out using Eq. (594). We assume that the
electric field is radial and spherically symmetric, so
. Application of Gauss' law,

(602) |

for , and

for . Note that the electric field generated outside the charge distribution is the same as that of a point charge located at the origin, . Equations (594), (603), and (604) yield

(605) |

(606) |

The reason we have checked Eq. (594) so carefully is that on close inspection
it is found to be
inconsistent with Eq. (585), from which it was supposedly derived!
For instance, the energy given by Eq. (594) is manifestly positive definite, whereas
the energy given by Eq. (585) can be negative (it is certainly negative for
a collection of two point charges of opposite sign). The
inconsistency was introduced into our analysis when we replaced Eq. (586) by
Eq. (588). In Eq. (586), the self-interaction of the th charge with its
own electric field is specifically excluded, whereas it is included in Eq. (588). Thus,
the potential energies
(585) and (594) are different, because in the former we start from
ready-made point charges, whereas in the latter we build up the whole
charge distribution from scratch. Thus, if we were to work out the
potential energy of a point charge distribution using Eq. (594)
we would obtain the energy (585) *plus* the energy required to assemble the
point charges. What is the energy required to assemble a point charge?
In fact, it is *infinite*. To see this, let us suppose, for the sake of argument, that
our point charges are actually made of charge uniformly distributed over a small
sphere of radius . According to Eq. (601), the energy required to assemble the
th point charge is

(607) |

(608) |