For starters, if we take a plate capacitor with length and width far larger than the distance between the plates and no electrical connection between the plates, the electrostatic field strength between the plates is independent of the distance between the plates and depends only on the charge per area on the plates. So, pulling the plates apart needs energy and this energy is stored in the electric field, the strength of which remains constant (and therefore the electric energy density) and the volume of which increases.

Now on to my question:

Imagine a flat and long coil (to be specific: lets assume the cross-section is a rectangle in the x-z-plane with width (x-direction) far bigger than height (z-direction), the axis of the coil is in y-direction, and the length is also far bigger that the height). The magnetic field energy density is independent of the height.

Now the parallel wires in x-directions (current flows in positive x-direction for the part with z>0 and in negative x-direction for the part with z<0) repel each other due to Ampere’s force law, so they move apart by this force and create a higher volume filled with the same magnetic field energy density and so the total energy increases (at first glance for free).

Is my conjecture correct that the energy comes from the voltage source, because in the wires an electric field is induced due to the movement of the wires in the magnetic field and therefore, to maintain the current, the voltage source has to supply energy? Can this be made quantitative?

Remark 1: In order to be able to pull in z-direction, one has either to assume that the electric wire is elastic, or one can imagine that the coil does not consist of one wire but of horizontal pieces of wire (x-direction) and that they are touching electrical rails right and left, where they can glide without resistance vertically.

Remark 2: I first thought that the external magnetic field (outside the coil) is essential, but I do no longer think so, as the same effect happens for a solenoid where the outer field is negligible for lengths far bigger than the radius and the magnetic field strength is independent of the coil radius and there is a radially outward pointing force to the wires. So imaging the wires being elastic the coil would “blow up” and thus the field energy would increase.