Linear dependence of magnetic potential on current density

Question

I'm a mathematician learning physics to provide some background for my mathematical work (especially pde's!). I have been reading through Jackson's Classical Electrodynamics (3rd edition), and I was puzzled by an assumption he makes. On page 214, he derives the equation $W=\frac12 \int_{V_1} \mathbf{J} \cdot \mathbf{A}$ under the assumption that the magnetic potential $\mathbf{A}$ and the charge density $\mathbf{J}$ are related linearly. However, this seems to be a very strict condition, since they are related in each coordinate by the Poisson equation. It seems like only eigenvalues of the Poisson equation could satisfy the linearity condition (after diagonalizing the linear relationship). And yet on the very next page, he uses the above formula for work in the very general setting of a system of $N$ arbitrary circuits.

So, my question is, how common is a linear relationship between the vector magnetic potential and the current density? And do Jackson's results hold in the settings he uses them in?

Answer 1

The essential physics this encodes is the superposition principle, which is at the heart of classical electromagnetic theory. What this states is that the fields from a collection of sources is the vector sum of the fields created by each different source. In particular this means that twice the currents generates twice the vector potential and twice the magnetic field, and so on, which boils down to a linear relation between the potentials and fields and the sources that generate them.

There is plenty of relatively direct experimental evidence for the superposition principle, but I think the consensus is that it is such a basic element of the theory that it is an essential postulate of its own, and that its validity should be shown in the overall success of the theory. Electromagnetism simply couldn't exist in its current form without it. Jackson discusses this in detail in pp. 9-13 along with the circumstances (pretty extreme in macroscopic terms) in which it could break.

Regarding your comments on the Poisson equation, my first reaction is that you should be careful since that equation is only valid in the Coulomb gauge and the gauge freedom should be handled with some care. I think, however, that your doubts are best addressed on the electrostatic analogue with the electric charge density and the electrostatic potential. These obey the simple scalar Poisson equation $$\nabla^2\varphi=\rho/\epsilon_0.$$ This is a linear equation (with no gauge uncertainty in it), and like all linear equations its solutions can be expressed as $$ \varphi=\frac{1}{\nabla^2}\rho/\epsilon_0+\varphi_0\textrm{, where }\nabla^2\varphi_0=0. $$ Here $\frac{1}{\nabla^2}$ is a nonlocal integral operator, which is really immaterial. $\varphi_0$ is a solution of the homogeneous equation and physically represents the potential of externally applied fields; for that physical reason it is always imposed to be constant (if not zero) for isolated systems. What this inverse equation then states is the superposition principle - sum of sources gives sum of potentials - in the shape of the mathematical result that the inverse of a linear operator (with the necessary uniqueness caveats) is always a linear operator.