The A E Phi's of Electromagnetism:

Taking Theory Back to the Source(s)

Daniel M. Dobkin
May 2007 revised Dec 2012

It’s Carver Mead’s Fault

Back around 2001, just as the first Internet boom was being fiberstrated, an old colleague (Alain Harrus) introduced me to Carver Mead's book, Collective Electrodynamics. It's a funny book, a mixture of a bit of history of science, some simple computational examples, and some arguments about things that are fun to philosophize over but not very important for practicing engineers. But the heart of the book is a quest to undo some choices made not so much by Maxwell as by Heaviside back in the late 18th Century: to emphasize the potential functions Phi and A, coupled intimately to the charges that are their sources, and abandon the derived quantities B and E as unphysical.

Ever since reading the book I've spent an unjustifiable amount of time trying to turn Mead's ideas into specific useful computational approaches for the kinds of problems engineers try to solve. I learned my electromagnetic theory from Richard Feynman's Lectures on Physics, Volume II, in which Feynman alludes to the importance of the vector potential A, but mostly approaches electromagnetism in the standard fashion, albeit with the inimitable Feynman intuition always in evidence. It was often challenging to rethink problems, and at times seemed pointless, but after some travail I have found that there are real benefits to be gained from a change in approach. Here's what I've learned:

B is a luxury, A a necessity.

Did you ever wonder why you take the curl and then the cross-product and somehow end up back in the direction you started? The underlying reason is that you didn't need to bother: in problems that involve straight wires or flat surfaces, B isn't needed and finding it is just a distraction. You just calculate A and derive the contribution to the electric field, -dA/dt, from it. For example, the inductance of a straight wire is much simpler to find directly from the vector potential than from the strangely indirect approach of calculating A and rederiving the force on the current from it, or the even more obscure derivation of the energy out to infinite. Curls and cross-products become dispensable in many cases.

In problems that involve coils of wire, the path integral of A is -- guess what? -- the same as an area integral of B! (This is a consequence of Stokes' theorem and the definition of the magnetic field.) So you can use either one. Sometimes (as in the case of a solenoidal coil) it can be much easier to find B and derive A than the other way around. However, even in the case of a solenoid, A is the real deal and B a convenience, as we'll show momentarily.

Real Sources Are All You Need

Currents launch a vector potential and charges launch a scalar potential. (In relativistic terms this is all part of a single four-vector, but I'm old and move pretty slow compared to c these days.) And that's all you need to know.

Okay, not really. You need the distribution of the currents, which is usually determined self-consistently from a requirement on the resulting potentials (such as zero electric field inside a metal). But you don't need to treat electric fields as sources, or use wavefronts as sources (Huyghens' principle), or introduce equivalent magnetic currents. It makes for a conceptually cleaner, simpler world view, though whether it is easier to calculate what you need to know from the real sources rather than the equivalent ones is rather dependent on the problem.

Another nice simplification that results from focusing on the sources is the elimination of artificial distinctions between "near" and "far" field. Rather than some mysterious self-sustaining interaction between electric and magnetic fields zooming off into space, we arrive at a much more direct view of radiation:

Everything radiates but most things cancel.

The near field is a mathematical distinction -- whether you need to account for variations in the inverse distance (1/r) across the source or not -- but not a change in the physics. Radiation resistance can be found locally (which is also true in standard theory, just not as obvious) rather than through an integration at infinite, although in doing so we use retarded potentials only, thus making an assertion about the remainder of the universe.

In the remainder of this discussion I'll try to demonstrate each of the points above through examples of real problems of interest to engineers where computations based on physical sources and the resulting potentials produce the correct answers, and are either simpler to implement, or offer alternative insights obscured by conventional methods. I don't think you're going to throw away your E&M texts or antenna books when you're finished, but I do think you'll find some obscure points clarified and easier ways of estimating things you need to know.