3 current loops with non-conservation of momentum, when one specifically considers SIGNs

Question

Thought experiment I.

Consider two loops of wire, 2 small dipoles B and C , with a common axis z (facing each other) and (say) 30 cm apart B to C. At the speed of light, information (including a change in magnetic field) will require 1 nanosecond to travel from C to B.

1. Have the current on in loop B for some period of time at the start, so the B-fields at C is established in the +z-direction.

2. Turn loop B off rapidly (fall time < 0.1 ns, say) at the same time that a current in loop C is turned ON (rapidly, rise time <0.1ns, and opposite sense with respect to the previous current in loop B ).

3. In this way, as the current is turned on in loop C, it is immersed in the field from loop B and therefore both receives an impulse to the right, in the +z-direction.

However, loop B will be "off" (and open so no effective eddy currents) when the "return" field from loop C arrives.

Thus, loop C (which is free to move) will experience an impulse giving it momentum in the +z direction (to the right), whereas loop B will not experience an impulse to the left.

I think this argument is sufficiently simple to sketch and to ponder.

Thought Experiment II. However, If you argue that there is momentum to the left "in the magnetic field" from loop B, I will add a third loop to the left (call it A), and again, as B is opened rapidly (short fall time) -- at the same time that a current in A is turned ON (rapidly, and SAME sense with respect to the previous current in loop B ).

You see, while one may hand-wave (without equations and ignoring signs) that a magnetic field "carries momentum", the fact is that one can choose the DIRECTION of the current in a loop (or loops) which absorbs energy from that field, and *thus one can choose the DIRECTION of the momentum in the sensing loop.*

In this way, loops A and C (both free to move) as they turn on are immersed in the field from B while having currents in the opposite sense (A attracted by loop B whereas C is repelled) -- therefore BOTH loops receive an impulse to the right, in the +z-direction.

Oh, and I will need to turn off the currents in loops B and C rather quickly, so that they both receive impulses in the +z direction without "feeling" the B fields from each other, for they will be "off" when those fields arrive.

If you're concerned about fringe fields, I can add a rod of very high magnetic permeability down the z-axis, extending from A to C, so that essentially all the magnetic field is contained on the z-axis.

Whew -- simple thought experiment, but one that could actually be done IMO.

What will happen? Will there be detectable momentum imparted to the right (+z direction), but not to the left?

Answer 1

I can see you've added more twist to your previous thought experiment, but it can be understood using a simple analogy. Think of the EM field as water, and the coils as boats floating in water. At the moment you turn on the electric current in the loop, the boat starts its propeller, and accelerates in +/- z direction (according to the direction of the current) by transferring momentum to the ambient EM field. So there's no need for the EM field to carry momentum in a predetermined direction. In fact, before you switch on the second coil, you have a static magnetic field (generated by the first coil) which has zero momentum density at every point. (EM momentum density = Poynting vector / c^2, therefore static E/B field doesn't carry momentum.)