Let us return to the equations of motion:

(8.55) | |||

(8.56) |

There is another way to make at least one of these two equations
simplify. We can just insist that:

(8.57) |

(8.58) |

As before, we propose:

(8.59) | |||

(8.60) |

such that

(8.61) |

(8.62) |

(8.63) |

If we use the Coulomb gauge condition (which we are now justified in
doing, as we know that the resulting potentials will lead to the same
physical field) the potentials in the Coulomb gauge must satisfy the
equations of motion:

(8.64) | |||

(8.65) |

The potential is therefore the

(8.66) |

(8.67) |

(8.68) |

In this equation one uses the value of the charge density on all space as a function of time under the integral, and then adds a source term to the current density in the inhomogeneous wave equations for the vector potential derived from that density as well.

There are several very, very odd things about this solution. One is
that the Coulomb potential is *instantaneous* - changes in the
charge distribution *instantly* appear in its electric potential
throughout all space. This appears to violate causality, and is
definitely not what is physically observed. Is this a problem?

The answer is, no. If one works very long and tediously (as you will,
for your homework) one can show that the current density can be
decomposed into two pieces - a *longitudinal* (non-rotational) one
and a *transverse* (rotational) one:

(8.69) |

(8.70) | |||

(8.71) |

Evaluating these pieces is fairly straightforward. Start with:

(8.72) |

(8.73) | |||

(8.74) |

(which are both Poisson equations).

With a *bit* of work - some integration by parts to move the
's out of the integrals which imposes the constraint that
and have compact support so one can ignore the surface term -
the decomposed currents are:

(8.75) | |||

(8.76) |

Substituting and comparing we note:

(8.77) |

(8.78) |

In the Coulomb gauge, then, only the *transverse current* gives rise
to the vector potential, which behaves like a wave. Hence the other
common name for the gauge, the *transverse* gauge. It is also
sometimes called the ``radiation gauge'' as only transverse currents
give rise to purely transverse radiation fields far from the sources,
with the static potential present but not giving rise to radiation.

Given all the ugliness above, why use the Coulomb gauge at all? There
are a couple of reasons. First of all the actual equations of motion
that must be solved are simple enough once one decomposes the current.
Second of all, when computing the fields in free space where there are
no sources, and we can find both
and
from
alone:

(8.79) | |||

(8.80) |

The last oddity about this gauge is that it can be shown - if one works
very hard - that it preserves causality. The transverse current above
is *not localized within the support of
* but extends
throughout all space just as instantaneously as does. One part
of the *field* evaluated from the solution to the differential
equations for
, then, must *cancel* the instantaneous Coulomb
field and leave one with only the usual propagating electomagnetic
field. This is left as a homework problem.