# 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?

• Any change to the field will propegate at the speed of light. The instantaneous change is OK as an approximation. Take a look at an article that describes Purcell's treatment of electric and magnetic fields based on Relativity. physics.weber.edu/schroeder/mrr/MRRtalk.html – Peter R Jun 24 '16 at 18:02
• I understand that the equation gives good results compared to what can be measured in the scale of a laboratory. I am wondering about what the equations say in principle and how we could write integral forms that would be suitable not just to lab scales, but to arbitrarily large scales. Thanks. I will take a look at the link. Sounds interesting. – bob.sacamento Jun 24 '16 at 18:05
• The induced EMF formula only holds in the quasi-static approximation, i.e. for slowly changing fields, which for any given rate of change can always be made correct for sufficiently small surface elements. For finite geometries the precision of the approximation depends on the chosen time scale. When it doesn't hold we have to solve Maxwell's equations in full, which, to use a euphemism, is "hard". – CuriousOne Jun 24 '16 at 18:45
• @CuriousOne, which induced EMF formula only holds in quasistatic approximations? All formulae in the question are equivalent to Faraday's law, i.e. they have no known violation. – Ján Lalinský Jan 5 at 2:53

## 1 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.

• OK, I get the point about the "up and down additional magnetic field" adding to zero. But, dang, that seems coincidental! Are you sure there are no issues with simultaneity here? – bob.sacamento Jun 24 '16 at 19:49
• @bob.sacamento Yes, because you can zoom out and prove, once and for all, that Maxwell's equations obey relativity. Taking it the other way, this kind of "coincidence" helped motivate relativity in the first place. – knzhou Jun 24 '16 at 19:51
• If you haven't studied SR and E&M together, I can see how it looks unnatural. But the more you learn, the more natural it looks -- they're deeply linked, and there's no dial you can turn to destroy that link. – knzhou Jun 24 '16 at 19:53
• Did some thinking on a 1D analogous problem. Don't have to have oscillations. A small but widening area of flux moves out from the coil (2 current sheets in 1D) while a thin but growing area of flux grows within the coil, so that the overall flux is zero ... until you find yourself within the leading edge of the outwardly expanding exterior flux. So you are basically correct. Might be instructive to try the 2D calculation. I am trying to work up the nerve. – bob.sacamento Jul 7 '16 at 19:26