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Derivation of capacitance for two shells

  1. Feb 13, 2017 #1
    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. jcsd
  3. Feb 13, 2017 #2


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    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.
  4. Feb 14, 2017 #3

    rude man

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    Two shells at equal radii? So what are R and r? The capacitance between two shells of "equal radii" would be infinite ...
  5. Feb 14, 2017 #4
    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.
  6. Feb 14, 2017 #5

    rude man

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    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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