Question 14 of 311

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $\theta$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is $1.5g/cc,$ the dielectric constant of water will be ______.
(Take density of water $=1g/cc$)

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Answer: 3

In air:
$\tan \dfrac{\theta }{2}=\dfrac{F}{mg}=\dfrac{{q}^{2}}{4\pi {\epsilon }_{0}{r}^{2}mg}$
In water:
$\tan \dfrac{\theta }{2}=\dfrac{{F}^{'}}{m{g}^{'}}=\dfrac{{q}^{2}}{4\pi {\epsilon }_{0}{\epsilon }_{r}{r}^{2}{mg}_{eff}}$
Equate both equations, we get
${\epsilon }_{0}g={\epsilon }_{0}{\epsilon }_{r}g\left[1-\dfrac{1}{1.5}\right]$
$\Rightarrow {\epsilon }_{r}=3$