Continuing from where we reached by the end of the Hint 3: if you will be using the Ampère’s circuital law (the first option of Hint 1), the plan was to 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. Since the dipole fields vanish inversely proportionally to the distance cubed, the contribution of the semicircular part of this loop to the Ampère’s circuital law will vanish, leaving only the integral along the axis, from -\infty to +\infty. This integral can be separated into two parts: (a) the integral inside the sphere, from -R to +\infty; (b) the integral outside the sphere, from -\infty to -R, and +R to +\infty. Based on all the hints given thus far, the first integral can be expressed in terms of the H-field strength inside the sphere, and the second integral – in terms of the B-field strength near the outer surface of the ball, which can be related to the B-field strength inside the ball using the continuity conditions at the interface of magnetic materials. In such a way, the Ampère’s circuital law yields a relationship between the H and B fields inside the sphere which must be valid additionally to the B(H)-dependence of the hysteresis curve.

If you will be using the second option of Hint 1, the expression relating the magnetic field B inside the sphere to its magnetization M can be found as follows. As it is well known, the Maxwell equations for electric and magnetic fields are symmetric, except that there are no magnetic charges. However, in order to carry over the results obtained for electric field to magnetic field (and the other way around), it is convenient to assume that there are also magnetic charges q_m, sources of magnetic field lines, H=\frac {q_m}{4\pi r^2}, which make the Maxwell equations completely symmetric (if we ignore the minus signs such as the one by the Faraday’s law). Then, magnetic dipoles d_m can be still thought to be current loops (d_m=IA, where I is the current strength, and A – the surface area of the loop), but they can be simultaneously thought to be a pair of equal and opposite magnetic charges at distance l, d_m=q_ml. Now, the steps you need to take, are: (i) find the magnetic field inside a sphere filled with homogeneous magnetic charge density \rho_m using the Gauss theorem for the magnetic field; (ii) consider two spheres filled with magnetic charges of equal by modulus and opposite magnetic charge densities, and find the magnetic field in that region of space where the two spheres overlap using the superposition principle (the consequence of the linearity of the Maxwell equations); (iii) consider the process where the distance l between the two spheres goes to zero while the magnetic charge density goes to infinity (notice that this limit corresponds to the sphere being filled with a homogeneous magnetic dipole volume density M=l\rho_m).

*Please submit the solution to this problem via e-mail to physcs.cup@gmail.com.** The fifth and final hint will appear here on 2nd January 2022. 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, 2nd January 2022**.