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The Potential of a Disk With a Certain Charge Distribution

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Please see the attach file for the question taken from "Classical Electrodynamics 3rd by Jackson".

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Solution Summary

The solution shows in detail how to find the potential everywhere using Legendre expansion once we know the potential on the z axis.

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First, let's calculate the potential along the z-axis.
The charge surface density is given by:
(1.1)
Where K is some constant to be determined later.
The potential along the z axis is given by (in polar coordinates):
(1.2)
The angular integration just gives a factor of , so:
(1.3)
Setting
(1.4)
Therefore:

(1.5)
This is a conducting disk, hence the potential over the disk is constant V, specificaly ...

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