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Q.
A conducting sphere of radius $R$ carrying charge $Q$ lies inside an uncharged conducting shell of radius $2R$. If they are joined by a metal wire, the amount of heat that will be produced is
AIIMSAIIMS 2009Electrostatic Potential and Capacitance
Solution:
The capacitances of two are $C_{1} = 4\pi\varepsilon_{0} R$ and $C_{2} = 4\pi\varepsilon_{0}\left(2R\right)$.
Initial energy = $E_{i} = \frac{Q^{2}}{2C_{1}}$
Final energy = $E_{f} = \frac{Q^{2}}{2C_{2}}$
Heat produced = $E_{i} - E_{f}$
$= \frac{Q^{2}}{2}\left[\frac{1}{4\pi\varepsilon_{0}R}-\frac{1}{2\times4\pi\varepsilon_{0}R}\right]$
$= \frac{1}{4\pi\varepsilon_{0}}\cdot\frac{Q^{2}}{2R}\left[1-\frac{1}{2}\right]$
$= \frac{1}{4\pi\varepsilon_{0}}\cdot\frac{Q^{2}}{4R}$