# Derivation of capacitance for two shells

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1. Feb 13, 2017

### Stendhal

1. The problem statement, all variables and given/known data
A spherical capacitor is formed from two concentric, spherical, conducting shells separated by vacuum. The inner sphere has radius r the capacitance is C.

What is the outer radius R?

Already solved the problem, but I'm more wondering on how to derive the equation that I used.
2. Relevant equations
Capacitance for a solid inner sphere and outer shell is:

$$\frac {4πε} {\frac {1} {R} - \frac {1} {r}}$$

While for two shells at equal radii, the capacitance is:

$$\frac {4πε*r*R} {R - r}$$
3. The attempt at a solution
The first equation is simple to figure out, but I'm not really sure how and why making the problem into two shells causes that change.

2. Feb 13, 2017

### TSny

If you write $\frac{1}{R} - \frac{1}{r}$ as a single fraction, you will see that the two formulas are the same, except R is the smaller radius in the first formula while R is the larger radius in the second formula.

3. Feb 14, 2017

### rude man

Two shells at equal radii? So what are R and r? The capacitance between two shells of "equal radii" would be infinite ...

4. Feb 14, 2017

### Stendhal

I'm still not getting what you're saying there.

Whoops, that would change how the question works a lot. My bad, R is suppose to be greater than r.

5. Feb 14, 2017

### rude man

If R > r then your first formula in your first post gives negative capacitance so you know that must be wrong!
To correct it, swap r and R. Then you get your second formula! Capiche?