How to calculate magnetic force using differentiation of magnetic energy?

Question

I tried to calculate the magnetic force between two circular current loops using numeric integration and differentiation of magnetic energy: $$ \int d^3\vec{r} \frac{1}{2 \mu_0} \vec{B}^2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1).$$ But after repeated examination, the result still comes out to be essentially exactly the negative of what I calculated using the force law $$\vec{F}= \int I d\vec{l} \times \vec{B} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2) ,$$ where the magnetic field is got using the Biot–Savart law $$\vec{B}(\vec{r})=\frac{\mu_0}{4 \pi } \int d^3 \vec{r}' \frac{I d\vec{l} \times \vec{r}' }{|\vec{r}'|^3} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3) .$$

I mean, when I calculate the total magnetic energy of two circular loops with opposite current in a configuration like in the following figure, they should repel right? And thus the longer the distance between them the lower the total magnetic energy right? But my result is the longer the distance between them the higher the total magnetic energy. But what is even more strange is that the differentiation of the numerically calculated magnetic energy essentially exactly matches in magnitude the force calculated using the force law (which, of course, give a repulsive result).

I wonder why, or would someone please redo this calculation and see if I get it right or wrong.

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For convenience of possible recalculation, the following analytical formula may be useful.

For a circular loop with current I, radius R, the result is, $$B_z(\rho, z)= \frac{\mu_0 I}{2 \pi} \frac{1}{\sqrt{(\rho+R)^2+z^2}} [K(k'^2)+ \frac{R^2-\rho^2-z^2}{(R-\rho)^2+z^2} E(k'^2)],\,\,\,\,\,\,\,\,(4)$$. $$B_r(\rho, z)= \frac{\mu_0 I}{2 \pi} \frac{z}{\rho\sqrt{(\rho+R)^2+z^2}} [-K(k'^2)+ \frac{R^2+\rho^2+z^2}{(R-\rho)^2+z^2} E(k'^2)] ,\,\,\,\,\,\,\,\,(5)$$ $$k'=\sqrt{\frac{4 R \rho}{(\rho+R)^2+z^2}} ,\,\,\,\,\,\,\,\,(6)$$ I got this from https://summit.sfu.ca/_flysystem/fedora/2023-04/DanielCarletonFinalUGThesisEngSci2022.pdf except for some minor corrections. There seems to be some typo there. For equation (4) - (6) which I use I confirmed using integration using the Biot–Savart law equation (3).

This expression has singularity at the wire position ($\rho=R, z=0$). When using this to calculate magnetic energy (1) by integrations, the singularity remains. I made some handwavy estimation that when calculating the differentiation of the magnetic energy to get the force such singular regions actually do not give finite contribution. So when I actually perform the calculations, I simply exclude spartial regions within distance $\epsilon$ from the wire. $\epsilon$ is chosen to be much smaller than other length scales in the question. Since the final result matches the result got by the force law in magnitude so well, I tend to guess this estimation is valid.

Answer 1

You get higher energy with increasing distance, because you assume constant currents in the loops. That is the correct result, because the farther apart the loops are, the less their fields cancel each other. When the loops get closer to each other, due to opposite currents, total magnetic field in many regions decreases. In the limit where the loops align at the same position in space, magnetic field becomes zero everywhere, and thus magnetic energy drops to zero, the minimum possible value.

Real superconducting current loops do not maintain constant currents when they move away from each other. Instead, motion of the loop 1 in magnetic field of the other loop 2 causes motional EMF in the loop 1, which acts against the existing current in loop 1 and decreases it. So the currents get lower, partial magnetic fields of both loops get weaker and this makes it possible that total magnetic energy decreases, even though the partial fields are cancelling each other less. When the loops get closer, the opposite happens - currents increase enough so the magnetic energy increases.

Your calculation and its result are correct when there is something maintaining the currents in the loops at constant values, like some ideal current source in both loops. When the loops move away from each other, to maintain the current at the same value, the current source has to act on the current with additional electromotive force that counteracts the motional EMF. Motional EMF acts to decrease the current, the current source EMF acts to increase it, and when they exactly cancel each other, current can remain constant. Then, magnetic energy in the system can increase as the loops move away from each other. This additional magnetic energy comes from the current source, working on the current.