Two identically charged spheres are suspended by strings of equal length. The strings make an angle of `30^(@)` with each other. When suspended in a liquid of density `0.8 g// c c` the angle remains same. What is the dielectric constant of liquid. Density of sphere`=1.6 g//c c`

Explanation

Initially as the forces acting on each ball are tension T,
weight mg and electric force F, for its equilibrium along vertical,
`T cos theta = mg` …(A)
and along horizontal
`T sin theta = F` …(B)

Dividing Eqn. (B) by (A), we have
`tan theta = (F)/(mg)` ...(C )
When the balls are suspended in a liquid of density `sigma` and dielectric constant K, the electric force will become (1/K) times, i.e., F' = (F/K) while weight
`mg' = mg - F_(B) = mg - V sigma g`
[as `F_(B) = V sigma g`, where `sigma` is density of material of sphere]
i.e., `mg' = mg[1-(sigma)/(rho)]`
[as `V = (m)/(rho)`]
So for equilibrium of ball,
`tan theta' = (F')/(mg') = (F)/(Kmg[1- (sigma//rho)])` ...(D)
According to given information `theta' = theta`, so from equations (D) and (C ), we have
`K = (rho)/((rho - sigma)) = (16)/((1.6 - 0.8)) = 2`