## Homework for Static Fields

1. (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?

1. (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.

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

2. 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: $$W_{total}={1 \over 8 \pi\epsilon_0}{q^2 \over R}$$

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