Derivation of capacitance for two shells

[Stendhal]

Homework Statement

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.

Homework 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:

$$ \frac {4πε*r*R} {R - r}$$

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.

 

[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.

 

[rude man]

While for two shells at equal radii, the capacitance is:
$$ \frac {4πε*r*R} {R - r}$$

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

 

[Stendhal]

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.

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

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

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

 

[rude man]

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.

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?