Question 48 of 311

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $30^{\circ}$ with each other. When suspended in a liquid of density $0.8 \mathrm{~g} \mathrm{~cm}^{-3}$, the angle remains the same. If density of the material of the sphere is $16 \mathrm{~g} \mathrm{~cm}^{-3}$, the dielectric constant of the liquid is

A4B3C2D1

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Answer: (C) 2

From F.B.D of sphere, using Lami's theorem
$$\frac{\mathrm{F}}{\mathrm{mg}}=\tan \theta$$
when suspended in liquid, as $\theta$ remains same,
$\therefore \quad \frac{\mathrm{F}^{\prime}}{\mathrm{mg}\left(1-\frac{\rho}{\mathrm{d}}\right)}=\tan \theta$
using (i) and (ii)
$$\frac{F}{m g}=\frac{F^{\prime}}{m g\left(1-\frac{\rho}{d}\right)} \text { where, } F^{\prime}=\frac{F}{K}$$
$\therefore \quad \frac{\mathrm{F}}{\mathrm{mg}}=\frac{\mathrm{F}^{\prime}}{\mathrm{mg} \mathrm{K}\left(1-\frac{\rho}{\mathrm{d}}\right)}$
or $\quad \mathrm{K}=\frac{1}{1-\frac{\rho}{d}}=2$