# Linear dependence of magnetic potential on current density

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?

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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.