Faraday's law and a moving charge

Question

Suppose we have a moving point charge with respect to some conductor which has a shape of a closed loop (e.g. a loop of wire). From the conductor's point of view the moving charge has a magnetic field and since the particle is moving through space, the magnetic field changes in time. That means that the flux trough the conductor loop is changing, which should cause a current to flow. However from the point charge's point of view there is only a regular Coulomb field with no curl and therefore no ability to produce electromotive force. Where is my misunderstanding?

Answer 1

By EMF, here I will mean circulation of electric field over a closed path in space, at a single time.

In conductor's frame $S$, for a closed path $\gamma$, we have $$ \mathscr{E}(t) = \oint_{\gamma} \mathbf E(t) \cdot d\mathbf s. $$ In charge's frame $S'$, for a different closed path $\beta$, we have $$ \mathscr{E}'(t') = \oint_{\beta} \mathbf E'(t') \cdot d\mathbf s'. $$

These two EMFs are not, in general, the same, not even if the loop $\beta$ is the Lorentz-contracted $\gamma$, because both formulae refer to arbitrary single but unrelated times $t,t'$. Time $t$ in the conductor frame does not imply any single time $t'$ in the charge frame, unless we fix the position to a specific point; but the loop is bigger than a point, and a range of times is implied.

The short answer:

1. in charge's frame, EMF of charge's conservative field for any loop is zero, but this is not the total EMF (net EMF) acting on the current in the conductive loop, because there is a contribution due to conductor's electric field as well;

2. total EMF for a moving loop, as opposed to EMF for a stationary loop, is not, in general, easily related to current in the loop like we know from the Kirchhoff second circuital law (valid for stationary loops and induced electric field). This law is a generalization of experience with static or slow moving loops made of Ohmic conductor; it is expected to fail at some circumstances. Using it in a frame where the loop moves fast seems to be such a case. There isn't even a single current in the loop in this frame, as the Lorentz transformation will change its value differently for different elements, depending on direction of the current element vector in space.

The longer rest of the answer:

In order to relate electric field in charge's frame $\mathbf E'$ to current in the loop, we may try to re-write the Kirchhoff second circuital law, valid in the loop frame, in terms of $\mathbf E',\mathbf B'$ in the charge frame, using the Lorentz transformations.

Let $\gamma_\nu(a)$ be four-vector $(t,\mathbf s)$ where the cartesian coordinates $\mathbf s$ are of a point of the loop $\gamma$ in the loop frame, corresponding to an integration parameter $a$, going from 0 to 1; and $\beta_\nu(b)$ a similar function $(t',\mathbf s')$, but for coordinates of the point of loop $\beta$ in the charge frame.

Net EMF in the loop frame is $$ \mathscr{E}(t) = \oint_{\gamma} \mathbf E(\mathbf s,t)\cdot d\mathbf s = \int_{0}^1 F^{0\nu} \frac{d\gamma_\nu}{da}da. $$ We work on the expression and try to express it using primed quantities from charge's frame: $$ F^{0\nu}(\gamma) \frac{d\gamma_\nu}{da} = \Lambda^{0}_{\rho} F'^{\rho \nu}(\gamma') \frac{\Lambda^\sigma_\nu d\gamma'_\sigma}{da}. $$

Here, although $F^{0\nu}$ denotes values of electric field components at a single time $t$, and $\gamma_\nu$ refers to that same time as well, the quantities on the right-hand side like $F'^{\rho \nu}$ and $\gamma'_\sigma$ are at a time $t'$ that depends not merely on $t$, but also on $\mathbf s$, the spatial coordinates of the point of the loop in loop's frame. Thus when integrating, the right-hand side has electric field evaluated at different times, and this seems very unlikely to give the same integral as the standard circulation of electric field in charge's frame at any single time, which would look like

$$ \int_{0}^1 F'^{0 \nu}(\beta') \frac{d\beta'_\nu}{db}db. $$

So, beware of applying the circuital laws to fast moving loops, it isn't likely to work well. Maybe if the loop is moving non-relativistically, and change of EMF in time is slow enough, it can be shown that the difference between the EMFs calculated in these two frames is negligible, but I did not check yet.

Answer 2

"That means that the flux trough the conductor..." The magnetic field due to the moving charge is circular, so there is no magnetic flux through the coil, and no induced current.