Volume charge density is expressed as:

$\overline{)\begin{array}{rcl}{\mathbf{\rho}}& {\mathbf{=}}& \frac{\mathbf{Q}}{\mathbf{V}}\\ & {\mathbf{=}}& \frac{\mathbf{Q}}{{\displaystyle \frac{\mathbf{4}}{\mathbf{3}}}{\mathbf{\pi R}}^{\mathbf{3}}}\end{array}}$

Now, let's consider distance r < R within the sphere.

The electric field at this distance is given by:

$\overline{){\mathbf{E}}{\mathbf{=}}\frac{\mathbf{k}\mathbf{q}}{{\mathbf{r}}^{\mathbf{2}}}}$, where q is the total charge enclosed by the sphere of radius, r.

We now have:

$\begin{array}{rcl}\mathbf{q}& \mathbf{=}& \mathbf{\left(}\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{\pi r}}^{\mathbf{3}}\mathbf{\right)}\mathbf{\rho}\end{array}$

A sphere of radius R contains charge Q spread uniformly throughout its volume. Find an expression for the electrostatic energy contained within the sphere itself

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