Momentum Conservation With Electromagnetism

I recently found an interesting question while perusing some internet archives. From the author of "Casey and Andy," he asks:"Imagine an electro-magnet in space. It is very powerful and very large. In fact, it weighs 1000 kilograms and is able to make a noticeable magnetic field 10 light-seconds away. Interestingly enough, there is a lump of iron 5 light-seconds away, and it also weighs 1000kg. Turn that magnet on for one second. What happens? ...Will the magnet gain a velocity toward the iron? If so, did it acquire that velocity at the same time the iron did? ...If the magnet doesn't move, then [have you] applied a force to the magnet without a counter-force[?] ...You could argue that the magnet will move toward the iron 20 seconds after the magnet was turned on. In other words, the magnetic field moved out to the iron, then came back and pulled on the magnet. But that means the iron had a 10 second head start on the magnet. When they meet, they will cancel each other's velocities, but they will not be at their collective center of mass. They will meet closer to the magnet's starting point than the iron's starting point. In other words, the system's center of mass moved without force or counter-force." From what I've been able to research, gravitons and photons both travel at the speed of c. Electromagnetic fields such as light or magnetism are a function of photons. Thus, it seems like the lump of iron would accelerate first after 10 seconds when the magnetic wave reached it. Other than that, I'm not certain. My best guess is that since magnetism isn't reflected (as far as I know), the sheet of iron will move towards the electromagnet, impact it, and the resulting weight and motion will cancel itself out. Is that accurate, or what would actually happen?
- Jon W. (age 28)
near Rochester, MN, USA

Nice question. Let's pick an explicit reference frame, because what's simultaneous depends on our choice of frames, and we don't want to accidentally switch mid-stream. The center of mass would be a convenient frame here.

At t= 5s, we see some EM disturbance propagating from the magnet, and infer that it must have switched on at t=0 s. At t=15 s we see that the iron moved, and infer that it started to move at t=10 s. We see no motion of the magnet. So how can momentum have been conserved? 

The answer is that the electromagnetic field itself has momentum. The momentum density in any region of space is proportional to ExB, where E is the electric field and B is the magnetic field. You may wonder how electric fields got into the story, which was about magnets. Maxwell's equations describing the space/time dependences of electromagnetic fields say that the changing B field is accompanied by an E field. 

So the total momentum in our frame will remain zero. Without actually solving the equations, I suspect that the asymmetrical set-up will leave a little bit of EM field momentum propagating out into space, so that the net momentum of the magnet (and its batteries, etc.) and the iron will not quite add up to zero. They'll keep moving in the direction opposite to the leftover field momentum.

Mike W.

(published on 11/10/2017)