Derivation of capacitance for two shells

  • #1
Stendhal
24
1

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.
 
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  • #2
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
Stendhal said:
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 ...
 
  • #4
TSny said:
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.
rude man said:
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.
 
  • #5
Stendhal said:
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?
 
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