Uniformly magnetized sphere, calculate force between the hemispheres

[fluidistic]

Homework Statement

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.

Homework Equations

Hints: $\vec B_{\text{int}}=\frac{2}{3}\mu _0 M_0 \hat z$
$\vec H_{\text{int}}=-\frac{1}{3} M_0 \hat z$
$\Phi_{\text{ext}}=\frac{1}{3} M_0 R^3 \frac{\cos \theta}{r^2} \hat z$

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.

The element ij of the Maxwell stress tensor has units of momentum per unit of area times time and gives the flux of momentum parallel to the ith axis crossing a surface normal to the jth axis (in the negative direction) per unit of time.

These units can also be seen as units of force per unit of area (negative pressure), and the ij element of the tensor can also be interpreted as the force parallel to the ith axis suffered by a surface normal to the jth axis per unit of area. Indeed the diagonal elements give the https://www.physicsforums.com/javascript:void(0) (pulling) acting on a differential area element normal to the corresponding axis. Unlike forces due to the pressure of an ideal gas, an area element in the electromagnetic field also feels a force in a direction that is not normal to the element. This shear is given by the off-diagonal elements of the stress tensor.

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".

 

[Orodruin]

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!).