Apparent instantaneous EM propagation paradox in two concentric solenoids Assume there are two solenoids $S_1$ and $S_2$ sharing the same axis, one of radius $R_1$ and the other of radius $R_2 > R_1$ resp.
For the sake of simplicity, I will assume that $S_2$ reduces to a single loop inside a plane $P$, that $S_1$ is relatively long with respect to $R_2$, and that $P$ crosses it at the middle.
At time $t = 0$, $S_1$ is suddenly strongly energized, such that the current rises very abruptly. During the current rising, the central solenoid generates a varying magnetic field $B(t)$, which is strong inside the solenoid, and weak outside and across $P$ (because $S_1$ can be assimilated to an infinite solenoid, due to its relative length with respect to $R_2$).
So, I believe it can be admitted that the flux $\Phi(t)$ of the magnetic field through the surface delimited by the circle of radius $R_2$ is not null and non constant; it follows that during the time the current rises inside $S_1$, the electromotive potential inside the loop of radius $R_2$ is $$-{d\Phi\over dt} \not= 0.$$
The main point is that that this electromotive potential exists in the exterior loop $S_2$ from time $t=0$ (or very close to it), while the loop is distant from $S_1$ by a length $R_2 - R_1$. This violates the principle of non instantaneous propagation of signals.
 A: If you are careful about all the delays, you will not find a paradox.
For example, you strongly energize a long solenoid by applying a voltage to the ends of the wires. Perhaps you close a switch. Current begins to flow in the solenoid. It takes time for the signal to propagate down the long winding.
You are assuming the long inner solenoid can be treated as infinite. The length must be much longer than $R_2-R_1$.
A: If you think there's really a paradox, then in order to resolve it, you should ask yourself which assumptions are definitely true, so you can reconsider any remaining assumptions by process of elimination. The assumptions that changes in the electromagnetic field cannot propagate faster than light should be assumed to be true here, since it is widely held to be a law of nature. Maxwell's equations should also be assumed to be true. What does that leave us with?
The final assumption you made was that there is an interval of time after the current has been switched on, but before $t = (R_2 - R_1)/c$, where the magnetic field configuration has reached its steady state configuration (where there is a longitudinal magnetic field at $r < R_1$ with magnitude $\mu_0 n I$, whereas for $r > R_1$ the field vanishes). Then you applied the integral form of the Maxwell–Faraday law and reached an apparent paradox.
It is this final assumption that cannot be true. In fact, when you turn on the current in the inner solenoid, the non-constant current will act as a source of radiation, which will propagate outward at the speed of light. Jefimenko's equations tell us that the magnetic field at a given point is obtained not only by integrating contributions from the electric current at each point in space at the "retarded time" (the current time minus the length of time it would take a light signal to travel from the source point to the observation point) but also has contributions from the time derivative of the electric current at the retarded time.
This tells us that, despite your best effort to turn on the current in the inner solenoid quickly, in order to try to establish the steady state magnetic field configuration inside the inner solenoid long before $t = (R_2 - R_1)/c$ is reached, it will not be possible to establish the steady state magnetic field configuration in the region $R_1 < r < R_2$ before $t = (R_2 - R_1)/c$ as there will be radiation propagating through this region. While it would be very difficult to do the explicit calculation, I feel comfortable in concluding that if you did it, you would find that the magnetic flux through the outer solenoid would in fact vanish when $t < (R_2 - R_1)/c$, taking into account the radiation field.
