Liénard–Wiechert potentials: Local or Material derivatives?

[particlezoo]

If I took a charged particle and accelerated it, that acceleration would have an effect on charges potentials, allowing for the radiation of electromagnetic waves. This acceleration would be local to a point in spacetime and the observed potentials would depend on the frame of reference of the observer.

However, if I had a train of such accelerating particles such that they formed constant current and charge density distribution, then I could have a situation where the fields wouldn't change, and in principle, the system would not radiate.

I know I posted about something similar before, but I am still curious as to why in the case of continuum source distributions, it appears in some equations of interest that the only accelerations that seem to matter are the ones that alter the local field over time.

If we use the Lorenz gauge, it is clear from the Liénard–Wiechert potentials that for a source charge q which is accelerating that its vector potential would change a rate determined in part by its acceleration. However, none of this acceleration appears to be apparent in the equations describing the Liénard–Wiechert potentials for a static continuum charge/current distribution. Since streamlines are not necessarily pathlines, the local currents (which follow the streamlines) are not necessarily moving in the same direction as the charges themselves (which follow the pathlines). It would seem that to extend the Liénard–Wiechert potentials from the form of moving independent charges to continuum charge/current distributions requires that we consider accelerations of charges along pathlines which may exist even if the flow field is unchanging.

However, in the literature the partial derivative of the vector potential over time is ∂A/∂t, which depends on the current densities of the sources. And yet, if we were to take into account the contribution to the change in the vector potential due to each source charge density, shouldn't the superposition then reflect the material derivative associated with these moving sources (hence DA/Dt) rather than the sources' local derivative (hence instead of ∂A/∂t)?

- Kevin M.

 

[facenian]

May be the foot note in "5.2 Biot and Savart Law" in Jackson third Edition can help you

 

[vanhees71]

Well, the footnote is rather cryptic. The point is that for time-dependent sources (charge-current distributions) you have to take the retarded propagator to integrate the Maxwell equations. For the four-potential the equation is the wave equation (Heaviside-Lorentz units),
$$\Box A^{\mu}=\frac{1}{c} j^{\mu}.$$
The retarded solution, corresponding to outgoing waves from the sources, reads
$$A_{\text{ret}}^{\mu}(t,\vec{x})=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \frac{j^{\mu}(t-|\vec{x}-\vec{x}|/c,\vec{x}')}{4 \pi c |\vec{x}-\vec{x}'|}.$$
That's it! Of course, for time-independent sources you get back the Coulomb Law for $A^0$ (which becomes the electrostatic scalar potential), and the Biot-Savart Law for $\vec{A}$. In this case you get the results of the Coulomb gauge, because for time-independent potentials the Lorenz-gauge fixing condition becomes of course the Coulomb one.