Question 16 of 311

Two identical charged spheres are suspended by strings of equal lengths. The string make an angle of $37^{\circ}$ with each other. When suspended in a liquid of density $0.7g{cm}^{-3}$, the angle remains same. If density of material of the sphere is $1.4g{cm}^{-3}$, the dielectric constant of the liquid is _____ $\left(\tan 37^{\circ}=\dfrac{3}{4}\right)$

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Answer: 2

For first case:
$Tcos\theta =mg$ and
$Tsin\theta ={F}_{e}$
Therefore, $tan\theta =\dfrac{{F}_{e}}{mg}$
$tan\theta =\dfrac{{F}_{e}}{{\rho }_{B}Vg}....(i)$
For second case:
$tan\theta =\dfrac{\dfrac{{F}_{e}}{k}}{{\rho }_{B}Vg-{\rho }_{L}Vg}.....(ii)$
From Eq. (i) & (ii), we get
${\rho }_{B}Vg=\left({\rho }_{B}-{\rho }_{L}\right)kVg$
$\Rightarrow 1.4=0.7k$
$\Rightarrow k=2$