# Work, energy stored in solid sphere

mathnerd15

## 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?

## Answers and Replies

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.

mathnerd15
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:
$$U=\int_V E^2d\tau + \int_S VEda$$
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##