Consider a spherical shell of radius $R$ with a total charge $+ Q$ uniformly spread on its surface (centre of the shell lies at the origin $x=0$ ). Two point charges $+q$ and $-q$ are brought, one after the other from far away and placed at $x=-a / 2$ and $x=+a / 2( < R)$, respectively. Magnitude of the work done in this process is
KVPY 2014, Diffcult
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(c)
Work done in the process $=$ Potential energy of the system
Also from shell theorem, charge $Q$ on shell behaves as a point charge at centre. So, magnitude of work done is
$\left|U_{\text {system }}\right| =\left|U_{12}+U_{23}+U_{31}\right|$
$=\mid \frac{k Q q}{a / 2}+\frac{k q(-q)}{a}+\frac{k Q(-q)}{a / 2}$
$=\left|\frac{k}{a}\left(2 Q q-q^2-2 Q q\right)\right|$
$=\left|\frac{-q^2}{4 \pi \varepsilon_0 a}\right|=\frac{q^2}{4 \pi \varepsilon_0 a}$
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