Two identical charged spheres of material density `rho`, suspended from the same point by inextensible strings of equal length make an angle `theta` between the string. When suspended in a liquid of density `sigma` the angle `theta` remains the same. The dielectric constant K of the liquid is

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` ...(1)
and along horizontal
`T sin theta = F` ...(2)
Diving Eqn. (2) by (1) we, have
`tan theta =F/(mg)` ...(3)
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)])` ...(4)
According to given information `theta'=theta`, so from equations (4) and (3), we have :
`K=rho/((rho-sigma))=((1.6))/((1.6-0.8))=2`