# Uniformly magnetized sphere, calculate force between the hemispheres

1. Aug 5, 2015

### fluidistic

1. The problem statement, all variables and given/known data
A uniformly magnetized sphere of radius R has a magnetization $\vec M=M_0\hat z$. Calculate the force between the hemispheres whose contact surface is the zx plane. Indicate the direction of the force.

2. Relevant equations
Hints: $\vec B_{\text{int}}=\frac{2}{3}\mu _0 M_0 \hat z$
$\vec H_{\text{int}}=-\frac{1}{3} M_0 \hat z$
$\Phi_{\text{ext}}=\frac{1}{3} M_0 R^3 \frac{\cos \theta}{r^2} \hat z$
3. The attempt at a solution
Not sure. Maybe using Maxwell stress tensor? https://en.wikipedia.org/wiki/Maxwell_stress_tensor

So that $\sigma _{ij}=\frac{1}{4\pi}[E_iE_j+H_iH_j-\frac{1}{2}(E^2+H^2)\delta_{ij}]$.
In my case only $\sigma_{zz}\neq 0$. I get $\sigma_{zz}=\frac{M_0^2}{72\pi}$.
But I am utterly confused on what this value represents.
Bold emphasis mine. So apparently I get a force orthogonal to the zx plane? But I don't see any differential element, so how do I integrate this to get the total force?
Also something strange is that I didn't most of the formulae given as "hints".

Last edited by a moderator: May 7, 2017
2. Aug 8, 2015

### Orodruin

Staff Emeritus
You have to define the area over which you want to compute the force. Then you can integrate the total contribution to the force by integration over the area. In this case, your area should be all of the zx-plane.

Your conclusion that only the zz component of the stress tensor is zero is also incorrect (note the appearance of the Kronecker delta in the last term!).