# Electrostatic Energy of a thin spherical shell

1. Aug 18, 2008

### wbrigg

hey,

i have a question from an exam paper which isn't worded too nicely (most of the questions on the exam are worded in similar ways :grumpy:)

The way i've done it is to first put in my first shell of infintesimal charge and then treat it as a point charge. Then i treat the next shell as just a point, and place it at a distance R from the point (electrostatic energy of this;
$$U = \frac{n dq^{2}}{4 \pi \epsilon _{0} R}$$

so the total Electrostatic energy of the whole thing (with all the shells assembled) is going to be:

$$U=\Sigma^{n}_{i=1} \frac{i dq^{2}}{4 \pi \epsilon_{0} R}$$ Where $$n dq = Q$$

I can turn that into an integral (Replacing one of the dqs with $$\frac{q}{dq}$$) ;

$$U = \int ^{Q}_{0} \frac{q dq}{4 \pi \epsilon_{0} R} dq$$

This goes to $$U=\frac{Q^{2}}{8 \pi \epsilon_{0} R}$$

now i make it so it's shells rather than points;

$$U=\int^{2\pi}_{0}\int^{\pi}_{0}\frac{Q^{2}}{8 \pi \epsilon_{0} R} d\theta d\varphi$$

$$U=\frac{\pi Q^{2}}{4 \epsilon_0 R}$$

Could someone confirm whether or not this is correct please?

Last edited: Aug 18, 2008
2. Aug 19, 2008

### weejee

Why don't you distribute your 'infinitesimal charge' equally over the spherical shell. Then, the electrinc field generated by it outside the shell is the same as when it is concentrated in the center.

The answer is U=Q^2/(8 pi epsilon R), as you obtained in the middle of your calculation. The reasoning is the same as yours. Though I can't understand what you mean by replacing a 'dq' with 'q/dq'. I think it should be replacing 'idq' with 'q'.

Just discard the last part - integrating over angles.

3. Aug 19, 2008

### Defennder

It should be possible to do this question by integrating the square of the magnitude of the E-field over all space, isn't it? The hint given doesn't cover this possibility. It seems easier to do this since Gauss law allows you to exploit symmetry to get E-field.

4. Aug 19, 2008

### wbrigg

Yeah, i meant replace the i with a q/dq - which is what you said. never used latex before, so it's kinda hard to skim through it and check everything's correct.

and it probably would be easier just to have integrated the field twice. :grumpy:

5. Aug 19, 2008

### wbrigg

thanks.

6. Oct 14, 2010

### ccheck13

So I know I'm bringing up an old one but I actually have this exact same problem, the only difference is its surrounded by a vaccume. What changes will that cause to the work/ answer?