A unit positive point charge is taken slowly through an… | NEET 2026 (Re-neet)

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Electric Potential & Equipotentials

A unit positive point charge is taken slowly through an infinitesimally thin tube that is inside a charged dielectric sphere of radius $R$ , having uniform positive charge density $\rho$ , as shown in the figure. The initial and final positions of the charge are marked by $A$ and $B$ at distance $2R$ and $3R$ respectively, from the centre of the sphere. In this process, the magnitude of the total work done on the point charge is $\frac{\rho R^2}{n\varepsilon_0}$ . The value of $n$ is: ( $\varepsilon_0$ is the permittivity of vacuum)

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Answer: (a)

Electric force is conservative, so work by the slow external agent $= q(V_B - V_A)$ for any path, including through the sphere.
Outside the sphere $V = \dfrac{Q}{4\pi\varepsilon_0 r}$ with $Q = \dfrac43\pi R^3\rho$ , so $V = \dfrac{\rho R^3}{3\varepsilon_0 r}$ .
$V_A = \dfrac{\rho R^2}{6\varepsilon_0}$ , $V_B = \dfrac{\rho R^2}{9\varepsilon_0}$
$|W| = \dfrac{\rho R^2}{\varepsilon_0}\left(\dfrac16 - \dfrac19\right) = \dfrac{\rho R^2}{18\varepsilon_0}$
So $n = 18$ , answer (a).

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