Constant magnetic field > The magnetic field of a moving charge. Induction of the magnetic field from a linear current > Problem 9.2.19

Statement

$9.2.19.$ a. Inside a large square circuit with current, many square micro-circuits with current are evenly distributed. The magnetic moment of each micro-circuit is $M_0$. Prove that at a distance much greater than the distance between the micro-contours, the induction of their magnetic field coincides with the induction of the magnetic field of a large contour, the magnetic moment of which is $nM_0$, where n is the
number of micro-contours inside the large contour.

b. A thin square plate with dimensions $a × a × h (h ≪ a)$ is magnetized in a direction perpendicular to
its plane. Magnetic field induction in the center of plate $B$. Determine the magnetic moment of the unit
volume of the plate substance.

Solution

$$ M = n \Delta S = nM_0$$

b)Find the equivalent current flowing along the edges of the plate:
$$MhS=IS$$
$$I = Mh$$
Here, the field in the center is represented as a superposition of four fields from the frame contour. A field from one frame:
$$dB_1 = \frac{\mu_0 Idlsin\varphi}{4\pi r^2}$$
$$dB_1 = \frac{\mu_0 I\frac{a/2d\varphi}{sin^2\varphi} sin\varphi}{4\pi \frac{(a/2)^2}{sin^2\varphi}}=\frac{\mu_0 Isin\varphi d\varphi}{4\pi a/2}$$
$$B_1 = 2\int_{0}^{\pi/4}\frac{\mu_0 Isin\varphi d\varphi}{4\pi a/2}$$
$$B_1 = \frac{\mu_0 I}{4\pi a/2}\cdot 2 cos(\pi/4)$$
From total 4:
$$B = 4B_1 = \frac{2\sqrt{2}\mu_0I}{\pi a}$$
Substitute the expression for the current and get the answer:
$$M = \frac{B\pi a}{\sqrt{8}\mu_0 h}$$

Answer

а)Q.E.D

б)$$M = \frac{B\pi a}{\sqrt{8}\mu_0 h}$$