Two identical charged spheres are suspended by string of equal lengths. The string make an angle of $37^{\circ}$ with each other. When suspended in a liquid of density $0.7 \mathrm{~g} / \mathrm{cm}^3$, the angle remains same. If density of material of the sphere is $1.4 \mathrm{~g} / \mathrm{cm}^3$, the dielectric constant of the liquid is_____$\left(\tan 37^{\circ}=\frac{3}{4}\right)$.
JEE MAIN 2024, Diffcult
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$T \cos \theta=\mathrm{mg}$
$\mathrm{T} \sin \theta=\mathrm{F}_{\mathrm{e}}$
$\tan \theta=\frac{\mathrm{Fe}_{\mathrm{e}}}{\mathrm{mg}}$
$\tan \theta=\frac{F_e}{\rho_B V g}$ $....(I)$
$\tan \theta=\frac{F_e}{\frac{k}{\left(\rho_B-\rho_{\mathrm{L}}\right) V g}}$ $....(ii)$
$\text { From Eq. (i) & (ii) }$
$\rho_{\mathrm{B}} \mathrm{Vg}=\left(\rho_{\mathrm{B}}-\rho_{\mathrm{L}}\right) \mathrm{kVg}$
$1.4=0.7 \mathrm{k}$
$\mathrm{k}=2$
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