A spherical capacitor consists of two concentric conducting shells of inner radius $a$ and outer radius $b$. Its capacitance is:
A$4\pi\varepsilon_0(a+b)$
B$4\pi\varepsilon_0\dfrac{ab}{b-a}$
C$\dfrac{4\pi\varepsilon_0(b-a)}{ab}$
D$\dfrac{2\pi\varepsilon_0}{\ln(b/a)}$
Answer & Solution
Correct answer: B. $4\pi\varepsilon_0\dfrac{ab}{b-a}$
Place charge $+Q$ on the inner shell, $-Q$ on the outer. Field between the shells is $E(r) = Q/(4\pi\varepsilon_0 r^2)$. Integrating gives $V = Q/(4\pi\varepsilon_0)\,(1/a - 1/b) = Q(b-a)/(4\pi\varepsilon_0 a b)$. Then $C = Q/V = 4\pi\varepsilon_0\,ab/(b-a)$. The $1/\ln(b/a)$ form is for a cylindrical capacitor.
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