# Problem 1 – Hint 5

Here is a step-by-step instruction what you need to do.
1. Show that for a spherical ball of homogeneous magnetization $M$, magnetic field inside the sphere is homogeneous, $\vec B=\hat z B_0$ , and outside the sphere is identical to the field of an ideal dipole (the field strength vanishes inversely proportionally to the distance cubed). Second half of Hint 4 explains how to do this.
2. Derive a linear relationship between $B$ and $H$ inside a spherical ball of homogeneous magnetization. This can be done by using Ampère’s circuital law, $\int_{-\infty}^{+\infty}H\mathrm dz= \int_{-\infty}^{-R}B/\mu_0\mathrm dz+ \int_{R}^{+\infty}B/\mu_0\mathrm dz + \int_{-R}^{+R}H\mathrm dz =0$ (see also the previous hints). However, it can be also done by using the relationship $B/\mu_0=M+H$, where $M$ is to be expressed in terms of $H$ using the relationship between $M$ and $H$, valid for the region inside spherical ball of homogeneous magnetization (for more details how to derive this linear relationship, see the second half of Hint 4). NB! When adding $M$ and $H$, pay attention to the signs!
3. Using the obtained relationship in conjunction with the provided $B(H)$-curve, find the $B$-field inside the sphere for the both temperatures.
4. Using the $B-M$ relationship obtained by pt 2, determine the magnetization inside the sphere for the both temperatures.
5. Based on the result of pt 4, determine the change of the magnetic dipole moment.

Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. There will be no more hints. The next updates of the intermediate results for the Problem 1 will be published after 13:00 GMT, 9th January 2022. The final deadline for the solutions of the Problem 1 is 13:00 GMT, 16th January 2022.