I am currently working through my Nuclear Physics question sheet, and one
part of the course has me stuck. Its just a small part, but how do you
show that the potential energy due to electrostatic forces of a uniformly
charge sphere of total charge Q and radius R is

3Q^2
U=----
20*Pi*e*R

Where e is epsillon nought.

I looked in the textbook "Modern Physics" and it just says that a simple
proof can show this, albeit in a slightly rejigged way with a factor of 3
fifths at the start.

Dave Hughes <david.hughes@hugheswalker.wanadoo.co.uk> wrote in
newsan.2004.12.07.19.14.43.215372@hugheswalker.w anadoo.co.uk:

one
uniformly
simple
3

Recognize that if you have a uniformly charged sphere of total charge Q,
radius r, and charge density p, and it has a potential energy X, then the
change in X from adding a thin shell of thickness dr is:

dX = 4*pi*r^2*p*(dr)*Q/(4*pi*e*r).

Then integrate from X=0 to X=R, using Q(r)=(4/3)*pi*r^3*p .

Eliminate p from the final answer by substituting Q(R).