I have been using Jefimenko's Equations for determining the electric and magnetic fields of a conductive coil with a (possibly changing) current. For most situations of interest this can be done to sufficient accuracy by approximating the wire as having zero radius and neglecting retardation. However, now I am investigating situations where a coil is in motion and am not sure of the correct approach: I would expect to see motional EMF in the form of an electric field when the coil is moving relative to my coordinate system, but cannot see how to calculate it.
Since we can assume the conductor has net zero charge, only the $\frac{\partial \vec{J}}{\partial t}$ term should come into play, giving the contribution to $\vec{E}$ from a point on a coil as $-K\frac{1}{|r|}\frac{\partial \vec{J}}{\partial t}$ (for $K$ a positive constant, $|r|$ the distance to the point, and $\vec{J}$ the current density at that point). Suppose we have a straight wire parallel to the $z$ axis and passing through $(1,0,0)$ through which passes current $\vec{J}$ in the $z+$ direction; the wire is moving in the $x+$ direction with velocity $v$ (very small compared to light-speed), and we are interested in the contribution to $\vec{E}$ at $(0,0,0)$ from an infinitesimal segment of wire intersecting the $xy$ plane.
How do we calculate this?
My Attempt So Far
Clearly $\vec{J}$ is everywhere $0$ outside the wire, and within the wire the $x+$ directed component of current created by the wire's motion vanishes since both the electrons and the protons make that same motion, so that $\vec{J}$ remains parallel to the $z$ axis just as in a stationary wire. Since $\vec{J}$ never changes direction, $\frac{\partial \vec{J}}{\partial t}$ must be parallel to it, i.e. it always points in the $z+$ or $z-$ direction and varies only in (signed) magnitude.
But now I cannot proceed to integrate $\frac{\partial\vec{J}}{\partial t}$ for a zero-radius wire; I cannot integrate it for any wire, because $\frac{\partial \vec{J}}{\partial t}$ can be nonzero only upon the wire's surface (everywhere else current is constant over a small enough time interval as the wire moves), but we have a discontinuity in current at the surface, so $\frac{\partial\vec{J}}{\partial t}$ cannot be defined. I can see that if I had a current distribution smoothly varying across a positive-radius wire's cross-section then $\frac{\partial \vec{J}}{\partial t}$ would on the right half of the wire be in the direction of current and on the left half opposite that direction, since it's moving to the right; the $-\frac{1}{|r|}$ factor would ensure a small net $\vec{E}$ at (0,0,0) in the direction of current, as the right-hand rule predicts, which seems promising.
But surely it is not necessary to derive some sort of complicated, continuous current distribution! Non-Jefimenko methods allow treatment of zero-radius wires just fine, since there is no difficulty in calculating the magnetic field; if my thinking above is in the right direction, the result must be independent of the distribution in the limit as radius goes to 0. In any case, a constant current should distribute uniformly, which doesn't fix the discontinuity issue. How can I solve this? Am I neglecting some important effect of retardation, or some Lorentz transform madness, that matters even at very small $v$? Or is there just some clever trick for dealing with the surface discontinuity that can be followed by a limit to zero radius? There must be a simple formula that comes out of this without the need to introduce limits of made-up distributions for every wire. Or is the entire approach misguided, and even with Jefimenko I need to "manually" and inelegantly calculate induced current from the changing $\vec{B}$ field where no $\vec{E}$ field exists? As you can see, I'm at a bit of a loss here.
