Definition of Capacitance

1. Feb 13, 2012

MarkovMarakov

1. The problem statement, all variables and given/known data
Hi, I would really appreciate clarification on the definition of capacitance in this context: Suppose I have 3 concentric metal spherical shells and they have charges Q1,Q2,Q3 and potentials 0,V,0 and radii a,b,c respectively, what is the capacitance of the configuration?

2. Relevant equations
C=Q/V

3. The attempt at a solution
I know what the capacitance is with only 2 spheres but I am not sure what it means with 3.

2. Feb 13, 2012

MarkovMarakov

Don't worry, I have figured it out! :)

3. Feb 13, 2012

gash789

I am inclined to answer although I am not 100% this solution is correct so please tell me if this is wrong.

As I'm sure your aware we calculate the capacitance for two concentric circles by
$C=Q/V \; ,$
where from Gauss's law we draw a sphere around the smallest shell.
$V=\int_{a}^{b}E dr = \frac{Q_{enc}}{4 \pi \epsilon} \int_{a}^{b} r^{-2} dr =\frac{Q_{enc}}{4 \pi \epsilon} \left(\frac{1}{a}-\frac{1}{b} \right)$
Since we are explicitly given the charges on each sphere, I believe the enclosed charge should be that of the innermost sphere. Now here is where I am stuck, the charge described in the capacitance equation refers normally to have a charge q and -q on each side, I am therefore going to assume that Q=abs(Q1-Q2)/2. This is probably wrong but I can't see any other way.
$C_{a \rightarrow b}=\frac{2 \pi \epsilon |Q_{1}-Q_{2}|}{Q_{1}\left(\frac{1}{a}-\frac{1}{b} \right)}$

This result holds in the three sphere setup since we only considered the enclosed charge (from Gauss). Therefore this could be easily extended for the third shell giving a second capacitance seen between b and c.

Hope that helps, and isn't completely wrong!

4. Feb 13, 2012

rude man

What is the implication if the inner and outer shells have the same ("zero") potential?