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Q. The nucleus of an atom can be considered to be a charge of +Ze uniformly distributed throughout a sphere of radius a. Show that the potential energy of a nucleus due to its charge is (3Z^2.e^2)/(20.pi.epsilon-0.a). What would the potential energy be if the charge was spread uniformly over the surface of the nucleus.

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I used Gauss' law and spherical coordinates (we can ignore all components other than R due to symmetry) to give the electric field (radial component):

Er = +Ze/(4.pi.epsilon-0.a^2)

Energy Density = 1/2.Epsilon-0.E^2

But we want total energy, thus:

Energy = 1/2.Epsilon-0.E^2.Volume

Energy = (Z^2.e^2)/(24.pi.Epsilon-0.a)

However, this is clearly wrong. Could someone tell me what I'm doing wrong?! Thanks