Work, energy stored in solid sphere

[mathnerd15]

Homework Statement

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

Homework Equations

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?

 

[Simon Bridge]

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

 

[Simon Bridge]

OK... you seem to be using:
$$U=\int_V E^2d\tau + \int_S VEda$$
... it's a good idea to explain your process.

You ended up with: $$U=\frac{1}{2k}k^2q^2\left( \frac{1}{5R}+\frac{1}{R}\right)$$ ... where $k=1/4\pi\epsilon_0$

Your next step is to simplify the expression.
Did you have any other questions?

 

[paranormal]

I appreciate your attempt at finding the energy stored in a uniformly charged sphere. However, I would like to point out that your solution is not entirely correct. The correct equation for the energy stored in a uniformly charged sphere is given by W = (3/5)(k)(Q^2)/R, where k is the Coulomb constant and Q is the charge of the sphere. This equation can be derived using the formula for the potential energy of a point charge and integrating over the volume of the sphere.

To answer your question, the work in this case represents the effort needed to bring the whole solid sphere in from infinity by point charges. This also corresponds to the stored energy in the sphere, as energy is required to bring in the charges and overcome the repulsive forces between them.