The magnetic vector potential

(315) |

since the curl of a gradient is automatically zero. In fact, whenever we come across an irrotational vector field in physics we can always write it as the gradient of some scalar field. This is clearly a useful thing to do, since it enables us to replace a vector field by a much simpler scalar field. The quantity in the above equation is known as the

Magnetic fields generated by steady currents (and unsteady currents, for that matter)
satisfy

(317) |

since the divergence of a curl is automatically zero. In fact, whenever we come across a solenoidal vector field in physics we can always write it as the curl of some other vector field. This is not an obviously useful thing to do, however, since it only allows us to replace one vector field by another. Nevertheless, Eq. (318) is one of the most useful equations we shall come across in this lecture course. The quantity is known as the

We know from Helmholtz's theorem that a vector field is fully specified by
its divergence and its curl. The curl of the vector potential gives us the magnetic
field via Eq. (318). However, the divergence of has no physical
significance. In fact, we are completely free to choose
to
be whatever we like. Note that, according to Eq. (318), the magnetic field
is invariant under the transformation

leaves the electric field invariant in Eq. (316). The transformations (319) and (320) are examples of what mathematicians call

This particular choice is known as the

It is obvious that we can always add a constant to so as to make
it zero at infinity. But it is not at all obvious that we can always
perform a gauge transformation such as to make
zero.
Suppose that we have found some vector field whose curl gives the
magnetic field but whose divergence in non-zero. Let

(322) |

(323) |

Let us again consider an infinite straight wire directed along the -axis and
carrying a current . The magnetic field generated by such a wire is
written

(324) |

(325) |

Let us take the curl of Eq. (318). We find that

(326) |

(327) |

(328) | |||

(329) | |||

(330) |

But, this is just Poisson's equation three times over. We can immediately write the unique solutions to the above equations:

(331) | |||

(332) | |||

(333) |

These solutions can be recombined to form a single vector solution

Of course, we have seen a equation like this before:

Equations (334) and (335) are the unique solutions (given the arbitrary choice of gauge) to the field equations (279)-(282): they specify the magnetic vector and electric scalar potentials generated by a set of stationary charges, of charge density , and a set of steady currents, of current density . Incidentally, we can prove that Eq. (334) satisfies the gauge condition by repeating the analysis of Eqs. (300)-(307) (with and ), and using the fact that for steady currents.