In class we did a pair of clicker questions involving "momentum of the EM fields", which was very puzzling. First, a STATIC capacitor/Solenoid combination (crossed E and B fields) has a nonzero ExB inside, and thus a nonzero "stored momentum" in the fields. It's not mechanical momentum, nothing is moving in this region, it's just E and B field that "have" momentum. But nothing is flowing, this seems rather counterintuitive.

Next, I suggested "shorting out" the capacitor. For a short while, current flows, and since it's in the presence of a B field, it feels a force. When we are done, the capacitor is discharged, no more stored momentum in the fields. But presumable the wire is now moving, so the wire now carries the momentum we started with.

The suggestion in class that I made was "see, momentum is conserved, all is good. We started with momentum in the field. We ended with momentum in the wire. No problem". Well, there is a problem! If this was all in a "closed box", I'm afraid something magical and unphysical has happened. Because the BOX is now moving. That cannot happen, without emission of electromagnetic radiation in one direction, you can't just have stuff happen inside a closed box and suddenly see the box start to move. That really DOES violate Newton's laws!

Note that I am skirting with some subtle issues.  The system viewed from outside as a “big box” starts with no momentum (the “center of energy” is initially at rest), and it has to end that way.   If you’re curious as to the resolution, you should read Babson et al (including Griffiths as 4th author), Am J Phys 77 (9) Sept 2009, p. 826, “Hidden momentum, Field momentum, and EM impulse”. They argue that such a simplistic discussion, ignoring fringe fields etc, “Sounds good, but it is almost entirely wrong”. I decided that for the pedagogical purposes in class, this would help us “make sense” of how momentum can be stored in static fields. But it's clear that there's some tricky ideas here. (Note also footnote 9 of this paper, which points out that Griffiths’ text and solution manual gets hidden momentum “almost entirely wrong” too! Apparently, one can quite successfully work with the stress tensor and field momentum without fully understanding all the subtleties of “hidden momentum”. And apparently, there is still publishable physics lurking in the pages of Griffiths textbook, >150 years past the days of Maxwell!