Work, energy stored in solid sphere

Homework Statement

Find the energy stored in a uniformly charged sphere of charge q, radius R

The Attempt at a Solution

$$Ein=\frac{qr}{4\pi\epsilon o R^3}, Eout=\frac{q}{4\pi\epsilon o r^2}... W=\int_{0}^ {R}\int_{0}^{2\pi}\int_{0}^{\pi}[\frac{qr}{4\pi\epsilon oR^3}] ^2sin\theta d\theta d\phi r^2\ dr+ \int_{R}^ {\infty }\int_{0}^{2\pi}\int_{0}^{\pi}[\frac{q}{4\pi\epsilon or^2}] ^2sin\theta d\theta d\phi r^2\ dr= 2\pi\epsilon o(\frac{q}{4\pi\epsilon o})^2(\frac{1}{5R}+\frac{1}{R})=\frac{q^2}{4\pi\epsilon o R}\frac{3}{5}$$

by the way the work in this case is also like the effort needed to bring the whole solid sphere in from infinity by point charges or also the stored energy?

The Attempt at a Solution

Simon Bridge
Homework Helper
Is this a uniforms sphere of charge (i.e. as you may find in an insulator), or a spherical shell of charge (i.e. the charges are being placed on a conductor)? It affects the integral.

But yep - the electrostatic energy stored in a system of charges is the work needed to assemble them from infinity.

Nice LaTeX ... you can make a newline with a \\ to avoid running off the end of the page;
you can make subscripts with _{} like this: ##\epsilon_0## and ##E_{out}##.
trig functions are written \sin \cos etc.

thanks! this is a solid sphere of charge. I'm curious how you develop the mathematics for this, is it based on a physical intuition or a mathematical result of the electric field equation

Last edited:
Simon Bridge
$$U=\int_V E^2d\tau + \int_S VEda$$
You ended up with: $$U=\frac{1}{2k}k^2q^2\left( \frac{1}{5R}+\frac{1}{R}\right)$$ ... where ##k=1/4\pi\epsilon_0##