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(A) The capacitance $C$ of a spherical capacitor with inner radius $a$ and outer radius $b$ is given by the formula $C = 4\pi \varepsilon_0 \frac{ab}{

Vedclass · Accepted Answer

$4\pi \varepsilon_0 \left( \frac{ab}{b - a} \right)$

Vedclass · Answer

(A) The capacitance $C$ of a spherical capacitor with inner radius $a$ and outer radius $b$ is given by the formula $C = 4\pi \varepsilon_0 \frac{ab}{b - a}$.Since shell $A$ is grounded,its potential $V_A = 0$.The potential of shell $B$ is $V_B = \frac{1}{4\pi \varepsilon_0} \frac{Q}{b} + \frac{1}{4\pi \varepsilon_0} \frac{q_A}{b}$,where $q_A$ is the induced charge on shell $A$.However,for a grounded inner shell,the capacitance is defined as the ratio of the charge on the outer shell to the potential difference between the shells.The potential difference is $V = V_B - V_A = \frac{Q}{4\pi \varepsilon_0} \left( \frac{1}{a} - \frac{1}{b} \right)$.Thus,$C = \frac{Q}{V} = \frac{4\pi \varepsilon_0}{\frac{1}{a} - \frac{1}{b}} = 4\pi \varepsilon_0 \frac{ab}{b - a}$.

Two concentric spherical shells $A$ and $B$ of radii $a$ and $b$ $(b > a)$ are placed in air. Shell $B$ is given a charge $+Q$ and shell $A$ is grounded. What is the capacitance of the system?

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