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 $0.8\,\text{g cm}^{-3}$, the angle remains the same. If density of the material of the sphere is $1.6\,\text{g cm}^{-3}$, the dielectric constant of the liquid is

JEE Main · 2010 · 11 Apr 2010 · Offline · Electrostatics

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of

3 0^{\circ}

with each other. When suspended in a liquid of density

0.8 g cm^{- 3}

, the angle remains the same. If density of the material of the sphere is

1.6 g cm^{- 3}

, the dielectric constant of the liquid is

Answer: (C)

On the sphere in air, Lami's theorem gives

\frac{F}{m g} = tan θ

. When the sphere hangs in the liquid, buoyancy reduces the effective weight to

m g (1 - \frac{ρ}{d})

and the medium reduces the Coulomb force to

F / K

, where

K

is the dielectric constant. Because the angle is unchanged,

\frac{F / K}{m g ( 1 - \frac{ρ}{d} )} = \frac{F}{m g}

, which gives

K = \frac{1}{1 - ρ / d}

. With liquid density

ρ = 0.8 g cm^{- 3}

and sphere density

d = 1.6 g cm^{- 3}

K = \frac{1}{1 - 0.5} = 2

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option