# Momentum Conservation With Electromagnetism

*Most recent answer: 11/10/2017*

- 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 **E**x**B**, 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)*