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Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 kg m ^{-3}$ and dielectric constant $21$ . If the angle between the strings remains the same after the immersion, then
$(A)$ electric force between the spheres remains unchanged
$(B)$ electric force between the spheres reduces
$(C)$ mass density of the spheres is $840 kg m ^{-3}$
$(D)$ the tension in the strings holding the spheres remains unchanged
$B,C$
$B,D$
$B,A$
$B,C,D$
Solution
In equilibrium
$T \cos \frac{\alpha}{2}= mg$
and $T \sin \frac{\alpha}{2}=F$
After immersed is dielectric liquid. As given no change in angle $\alpha$.
So $T \cos \frac{\alpha}{2}= mg – V \rho g$
when $\rho=800 Kg / m ^3$
and $T \sin \frac{\alpha}{2}=\frac{ F }{ e _{ r }}$
$\therefore \frac{ mg }{ F }=\frac{ mg – V \rho g}{\frac{ F }{ e _{ r }}}$
$\frac{1}{e_r}=1-\frac{\rho}{d}$
$d=\text { density of sphere }$
$\frac{1}{21}=1-\frac{800}{d}$
$d=840$
