Question

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

Answer 1

Consider the forces acting on the spheres in air and water:

$\tan \frac{\theta}{2} = \frac{F}{mg} = \frac{q^2}{4 \pi \epsilon_0 r^2 mg}$

When suspended in water:

$\tan \frac{\theta}{2} = \frac{F'}{mg'} = \frac{q^2}{4 \pi \epsilon_0 \epsilon_r r^2 m g_{\text{eff}}}$

Equating both expressions and using the relation:

$\epsilon_r g = \epsilon_0 \epsilon_r g \left[ 1 - \frac{1}{1.5} \right]$

Simplifying gives:

$\epsilon_r = 3$