Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of…
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
4
3
2
1
Solution
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 $\quad \mathrm{K}=\frac{1}{1-\frac{\rho}{d}}=2$
