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## Homework for Static Fields

- (RepulsionNSGEM238)
*Students calculate the force between two hemispheres of a metal sphere, from Griffiths E&M (Note: This problem might be better as a challenge problem).*A metal sphere of radius $R$ carries a total charge $Q$. What is the force of repulsion between the “northern” hemisphere and the “southern” hemisphere?

- (EnergyGEM234)
*Students calculate the energy of two charged spherical shells using two different methods, from Griffiths E&M. (Note: Students find this wording difficult and so this problem has not been used in recent years).*Consider two concentric spherical shells, of radii $a$ and $b$. Suppose the inner one carries a charge $q$, and the outer one a charge $-q$ (both of them uniformly distributed over the surface). Calculate the energy of this configuration.

Starting from: $$W= {\epsilon_0\over 2}\int_{\hbox{all space}}E^2 \, d\tau$$

Starting from: $$W= W_1 + W_2 + \epsilon_0\int_{\hbox{all space}}\left(\vec E_1\cdot\vec E_2\right)\, d\tau$$

and using the result that the total energy of a uniformly charged spherical shell of total charge $q$ and radius $R$ is: $$W_{total}={1 \over 8 \pi\epsilon_0}{q^2 \over R}$$