Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of `30^@` with each other. When suspended in a liquid of density `0.8gcm^-3`, the angle remains the same. If density of the material of the sphere is `1.6gcm^-3`, the dielectric constant of the liquid is

Options

A. 1
B. 4
C. 3
D. 2

Answer

D

Explanation

Before the spheres are dipped in a liquid , we can write the following equation for equilibrium :
`T_1 cos theta =mg`…(i)
`T_1 cos theta =F_"el"` ….(ii)

On dividing the above equations we can write the following :
`tan theta =F_"el"/"mg"` ....(iii)
When the spheres are dipped into a liquid then due to force of buoyancy effective weight charges to
`mg(1-"density of liquid"/"density of sphere")="mg"(1-0.8/0.6)=(mg)/2`
and if dielectric constant of the liquid is K then the force between the charges changes to `F_"el"/K`.
Let the tension changes to `T_2`. Now , we can write the following equations for the equilibrium of the spheres :
`T_2 cos theta =mg//2`...(iv)
`T_2 sin theta =F_"el"/K`...(v)
Dividing the above two equations we get the following :
`tan theta =(2F_"el")/(Kmg)` ...(vi)
On equating the results of equation (iii) and (vi) we get K=2