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RawrSpoon
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Homework Statement
Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω, and surface charge density σ. Use the Maxwell Stress Tensor
Homework Equations
[tex]F=\oint \limits_S \! \vec{T} \cdot da - \epsilon_0 \mu_0 \frac {d}{dt} \int \limits_V S d \tau[/tex]
Because we're dealing with steady currents, the second term goes to zero.
I know that with a sphere, we have
[tex]da=r^2 sin \theta d \theta d \phi[/tex]
I also know that
[tex]B= \begin{array}{11} \frac {2 \mu_0 \sigma R \omega}{3}\hat{z} & inside \\ \frac {2 \mu_0 m}{3 r^3}(2 cos \theta \hat{r} - sin \theta \hat{\theta}) & outside \end{array} [/tex]
where
[tex]m=\frac{4}{3} \pi \sigma \omega R^4[/tex]
The Attempt at a Solution
I think I find Tzz since Txz and Tyz would be 0 because Bx and By are zero?
I'm honestly very confused by tensors in general, I have a pretty good idea about what they do and how they work, but I don't really know how to work them myself. Please point me in the right direction, I'm very lost as to what to do and I don't want to seem like I'm being lazy but I've broken my head all day trying to figure out what to do. The solutions manual doesn't give me an answer that makes me feel like I understand what's going on, and I can't really find anywhere that breaks it down enough for me to get that aha moment where I get it.
Thank you very much for all help :)
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