Problem 1 – Hint 3

Based on Hint 2, you can be sure that the magnetization inside the sphere is homogeneous: the vectors M and B are constant.
If you will be using the Ampère’s circuital law (the first option of Hint 1), notice that outside the sphere, the magnetic field is the field of an ideal (i.e. geometrically infinitesimal) dipole. (Indeed, a constant M can be thought to be created by two spheres, each filled with homogeneous magnetic charge density \pm \rho, and displaced by M/\rho relative to each other. Then, the field outside of those spheres is the same as that of two magnetic point charges at the centers of the spheres; at the limit \rho\to\infty, the two balls merge into a single ball, and the field outside becomes the field of an ideal dipole.) Consider a loop that goes through the sphere along its axis, extends in both directions very far from the sphere, and closes onto itself by drawing a very big semicircle.

If you will be using the second option of Hint 1, find an expression relating the magnetic field B inside the sphere to its magnetization M

Please submit the solution to this problem via e-mail to The fourth and final hint will appear here on 26th December 2021. For full regulations, see the “Participate” tab. The next updates of the intermediate results for the Problem 1 will be published after 13:00 GMT, 26th December 2021.