Simple explanation to the induction from the slowly changing $\vec B$ of a solenoid in the region of $0$ magnetic field

Question

I would like to get some elementary intuition into the problem a solenoid fed with a time-dependent current, and the resulting current that such the solenoid field would induce in a loop completely outside the solenoid.

The classic situation is to imagine a perfect solenoid of $n_1$ turns per meter, having radius $a$ and fed by a current $I(t)=I_0e^{-t/\tau}$. One places $N_2$ coils (all having radius $R>a$) around the solenoid as illustrated by this picture from Haliday&Resnick 10th Edition.

Using the usual Ampère's law argument shows the $\vec B$ field of the solenoid would be completely contained inside the solenoid, and homogenous across the cross section of the solenoid.

If there is a slowly varying current in the solenoid, will there be an induced current?

Presumably the answer is yes (at least according the HR solution manual): if we take the $\vec B$ field of the solenoid to be changing with time as a result of the current changing in time, but $\vec B$ still homogenous across its cross-section and $0$ outside, there is a change in the flux of this field through the surface bounded by the coils, and even though there is no magnetic field outside the solenoid where the coils are located. The resulting change in flux induces an EMF independent of $R$, the radius of the coils, or for that matter the shape of the coil, provided it completely contains the solenoid.

If you are suspicious about the use of Ampère's law for slowly varying currents, the same solenoid field is obtained explicitly by Das Gupta in "Magnetic field due to a solenoid." American Journal of Physics 52 (1984): 258-258, starting from Biot-Savart, which certainly holds for quasi-static currents.

But how can one intuitively grasp that no field outside the solenoid induces a current in a stack of coils located completely outside this solenoid, in the region where $\vec B=0$?

A "classic" explanation might that the field might be $0$ but the vector potential is $\ne 0$, yet it seems this is invoking a lot of heavy machinery for a 1st year physics problem. Moreover, Aharonov-Bohm-like explanations are really quantum in their nature and show that in quantum mechanics the potentials are the essential quantities.

Nota: a possible path of solution would be to invoke hidden momentum, along the argument in this file by K.T. McDonald. Is there a simpler explanation?

Answer 1

There is a magnetic field outside the solenoid and as the current changes there is a change in the magnetic field strength both inside and outside the solenoid.

This animation from MIT shows the effect of increasing the speed of positive charges (current) in the coils and here are three stills from it.

I think the point is that the magnetic field which is "generated" as the current increases in "compressed" within the solenoid but spreads out outside the solenoid so this is where the idea of no magnetic field outside the solenoid comes from?
In this way anything outside the solenoid, eg your outer coil, will know what is going on inside the solenoid because of the flow of electromagnetic energy out of the coil.

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I am tempted to think of magnetic field lines in the form of loops passing through the inside and then outside of the solenoid being created and the part formed outside the solenoid flowing out from the solenoid?

Answer 2

There’s an azimuthal electric field outside of the solenoid. This electric field is what causes current to flow through the outer coils (e.g. as given by Ohm’s law $\bf{J} = \sigma \bf{E}$).

There's no single physical explanation for the origin of this external electric field; "why" questions in physics rarely have a single answer if there are multiple mathematical derivations of the same result, and it's often a matter of personal opinion which derivation is the most illuminating. Here are four possible arguments/explanations for this external electric field:

1. The integral form of Faraday's law. (This is the most mathematically straightforward derivation - it's essentially the same one that you laid out in your question, with the additional identification of the emf with $\oint {\bf E} \cdot d{\bf l}$ - but as you say, it doesn't provide much physical intuition because it isn't manifestly local.)
2. The differential forms of Faraday's law and Gauss's law for electricity require that outside of the solenoid, any electric field must be both curl-free and divergence-free. Zero electric field would trivially satisfy these requirements, but it wouldn't satisfy the boundary conditions at the solenoid: there's an azimuthal electric field inside of the solenoid induced locally by the changing internal magnetic field via Faraday's law, and this internal electric field is nonzero at the solenoid. And there's no net charge at the solenoid, so the electric field must be continuous across it, so it must be nonzero and azimuthal just outside of the solenoid and decay with distance away from it.
3. The magnetic vector potential changes over time outside of the solenoid, and so ${\bf E} = -\partial {\bf A}/ \partial t$ (using temporal gauge for simplicity) cannot be zero. (This explanation is not necessary for explaining this phenomenon, though.)
4. The Jefimenko's equation for ${\bf E}$ contains a term that in this case simplifies to $-\iint_S d^2 {\bf r}'\ \dot{\bf K}({\bf r}', t_r)/|{\bf r} - {\bf r}'|$, where ${\bf K}$ is the surface current at the solenoid and $t_r$ is the retarded time. Therefore the electric field can be thought of as being sourced by a changing current, and in this case the changing current through the solenoid induces an external electric field. (Note that Jefimenko's equations don't apply exactly in the quasistatic regime in which this effect was derived, but presumably a similar term is found in the quasistatic equivalent to Jefimenko's equations, which I suspect would be quite challenging to derive.)

The fourth explanation is the most physically direct (although the most mathematically complex) and so possibly the most satisfying, although making it completely rigorous would require a challenging calculation.

Answer 3

First of all, the magnetic field outside the solenoid is not zero, even in the static case, because the field lines running parallel to the axis inside must exit from the north pole and wrap around--outside--before entering the south pole. Nevertheless, at a given distance to the side, somewhere near the midplane of the solenoid, the longer the solenoid the smaller the field, because the field lines spread farther apart.

If the current starts increasing, the result cannot simply be an increasing version of the static field. That is, as the number of field lines increases within the the solenoid, those same (continuous, right?) field lines cannot instantly take their final places in the static field outside, because the fact that the current started increasing cannot be transmitted faster than the speed of light. So what does happen?

Because field lines are continuous loops, as new field lines move inward from the increasing cylindrical current wall, their completions must move outward from the wall. The outward moving field lines "cut" any wire loop around the solenoid, which is what induces the emf. As field lines are a representation of directed magnetic flux, another way to say it is that the emf is induced by outward moving flux. Emf induced by cutting magnetic field lines was well understood by Faraday and Maxwell, but the concept has in recent decades fallen prey to a dogma that has arisen against the idea.

The magnetic field due to the rate of change of current can be understood in terms of the Biot-Savart law. The key is that, since information only propagates at speed c, at a given distance outside the solenoid the current in the near side of a solenoidal turn of wire appears slightly different than on the far side. This difference goes away when the current is steady again. Then the field returns to a static configuration once the new field lines find their final places.