Monopole and Dipole Terms of Electric potential (V) on Half Disk

jkthejetplane
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Homework Statement
So this is from my last exam. I wanted to figure this out as we can get some credit back on a question plus finals are coming up.
Relevant Equations
On my attached attempt you can see the equations I use as well as a few of the professor's notes
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Calculate potential on the z axis in terms of a Legendre polynomials. Then the off axis expansion will follow.
 
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It's given a gas of particles all identical which has T fixed and spin S. Let's ##g(\epsilon)## the density of orbital states and ##g(\epsilon) = g_0## for ##\forall \epsilon \in [\epsilon_0, \epsilon_1]##, zero otherwise. How to compute the number of accessible quantum states of one particle? This is my attempt, and I suspect that is not good. Let S=0 and then bosons in a system. Simply, if we have the density of orbitals we have to integrate ##g(\epsilon)## and we have...
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