Constant magnetic field > The magnetic field of a moving charge. Induction of the magnetic field from a linear current > Problem 9.2.23

Statement

$9.2.23.$ The interaction force of two thin magnetized square plates located at a distance $H$ above each other is equal to $F$. The dimensions of the plates are $a × a × h$. Estimate the magnetic moment per unit volume of the plate if the thickness of the plate is $h ≪ H$ and $H ≪ a$.

Solution

Since $h<<H$ we will consider the interaction of two frames with a current $I = Mh$. So, seeing that $a>>H$, we will take into account the interaction of only the nearest beams, the force of interaction between them is calculated assuming that the field of rotation is close to the field of infinite filaments:

$$B = \frac{\mu_0 Mh}{2\pi H}$$
$$F_1 = \frac{\mu_0 (Mh)^2a}{2\pi H}$$
$$F = 4F_1 = \frac{2\mu_0 (Mh)^2a}{\pi H}$$
$$M = \sqrt{\frac{F\pi H}{2\mu_0 a h^2}}$$

Answer

$$M = \sqrt{\frac{F\pi H}{2\mu_0 a h^2}}$$