electromagnetism - A puzzler in induction and special relativity

Some reading I was doing jogged my memory about a puzzle in E&M that hit me back in my undergrad days, but I just let it go at the time and never found an answer.

A pretty common undergrad E&M problem goes something like, "We manage, by way of external coils, to produce a magnetic field going through the center of a wire loop. The field occupies and area of A within the loop and we make it change at x Tesla/sec. What is the EMF induced around the loop?"

And of course we solve this with Faraday's equation

$\displaystyle \nabla \times E = -\frac{\partial B}{\partial t}$

Integrate it over the surface contained by our wire loop and get

$\displaystyle EMF = -\frac{d \Phi}{dt} = -A\frac{dB}{dt}$

Straightforward. Except ....

The EMF, according to this equation, is directly related to the instantaneous rate of change of $B$. And this is true even if $B$ is confined to a very small area within the current loop. It is true even if, for instance, $B$ is confined to, say, one cm square, and the current loop is one million miles in radius. Again, the relationship is instantaneous. So, if we turn on our coils and generate $B$, immediately there is an EMF 1 million miles away. But information cannot travel faster than the speed of light, so this makes no sense. Where have we gone wrong?

I am pretty sure the answer to this question has something to do with the "slipperiness" of the concept of simultaneity in special relativity. So taking an integral across the entire area of the current loop "simultaneously" is probably a tricky thing to do. So maybe the question to be asked is, given the strictures of special relativity, can we cast the Maxwell's equations, particularly induction, in meaningful integral forms, and what do those forms look like?

Answer

Suppose, as you said, we have a thin solenoid pointing upward, going through the middle of a loop one light-year wide. Then the three equations $$\nabla \times E = - \frac{\partial B}{\partial t}, \quad \mathcal{E} = -\frac{d\Phi_B}{dt}, \quad B = \mu_0 n I$$ together imply violation of causality, as shown in your question.

Maxwell's equations are already fully relativistic, so the first two equations are always true. In particular, the path from equation (1) to equation (2) is pure math and always holds; there's no need to modify that for relativity either.

The mistake is in equation (3). This is the field of a solenoid that has been on for a long time, but you want us to consider the case where it's "suddenly" switched on. If we switch the solenoid on quickly, $\partial B / \partial t$ is large, so we suddenly get a large electric field circulating just outside the solenoid. But then we have to account for the other induction effect, $$\nabla \times B \sim \frac{\partial E}{\partial t}.$$ If you draw a picture, you'll see this produces both an upward and downward additional magnetic field. A detailed calculation shows that, if you take a light-year-wide loop, the total magnetic flux is zero! Then $\mathcal{E}$ is zero, as expected by causality.

This additional magnetic field will induce additional electric fields, which induce more magnetic fields, so that after a year, the fields propagate to the loop and $\mathcal{E}$ becomes nonzero. You might already know this propagation mechanism by a different name; it's called light.