A solid sphere of radius R has a uniform charge density and total charge Q. Find the total energy of the sphere U in terms of

Explanation:

From the information given;

The surface area of a sphere =

If the sphere is from the collection of spherical shells of infinitesimal thickness = dr

Then,

the volume of the  thickness and the sphere is;

V =

Using Gauss Law

here,

q(r) =charge built up contained in radius r

since we are talking about collections of spherical shells, to work required for the next spherical shell r +dr is

where;

dq which is the charge contained in the  next shell of charge

here dq = volume of the shell multiply by the density

equating it all together

Integration the work required from the initial radius r to the final radius R, we get;

Recall that:

the total charge on a sphere, i.e

Then :

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