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. 2
C. 3
D. 4

Answer

B

Explanation

Before the spheres are dipped in a liquid , we can write the following equations for equilibrium :
`T_1 cos theta =mg`…(i)
`T_1 sin theta =F_"el"` …(ii)
On dividing above equations we get : `tan theta =F_"el"/(mg)` ….(iii)
When the spheres are dipped into a liquid then , due to the force of buoyancy, the effective weight changes to `mg(1-"density of liquid"/"density to sphere")=mg(1-1/2)="mg"/2` and if dielectric constant of the liquid is K then the force between the charges changes to `F_"el"/K`.
If the tension changes to `T_2`, then 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 above two equations we get :
`tan theta =(2F_"el")/(Kmg)`...(vi)
On equating results of equation (iii) and (vi) we get K=2