Question

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of 37° with each other. When suspended in a liquid of density 0.7 g/cm³, the angle remains the same. If the density of the material of the sphere is 1.4 g/cm³, the dielectric constant of the liquid is ___________. (tan 37° =\( \frac{3}{4}\))

Answer 1

Correct Answer: 2

The gravitational force acting on the spheres is given by:

\( F_g = \rho_B V g \),

where \( \rho_B \) is the density of the spheres, \( V \) is the volume of the sphere, and \( g \) is the acceleration due to gravity.

The electric force acting between the spheres is:

\( F_e = \frac{Q^2}{4 \pi \epsilon_0 d^2} \),

where \( Q \) is the charge on the spheres, \( d \) is the distance between the centers of the spheres, and \( \epsilon_0 \) is the permittivity of free space.

The buoyant force acting on the spheres in the liquid is:

\( F_b = \rho_L V g \),

where \( \rho_L \) is the density of the liquid.

In equilibrium, the tension in the strings balances both the horizontal and vertical forces. The vertical and horizontal components of the tension are:

\( T \cos \theta = F_g - F_b \),

\( T \sin \theta = F_e \).

The ratio of these equations gives:

\( \tan \theta = \frac{F_e}{F_g - F_b} \).

For the spheres in the liquid, substituting the expressions for \( F_g \) and \( F_b \):

\( \tan \theta = \frac{F_e}{(\rho_B - \rho_L) V g} \).

When the spheres are in air, the buoyant force is negligible, and the force balance becomes:

\( \tan \theta = \frac{F_e}{\rho_B V g} \).

Equating the two expressions for \( \tan \theta \), we get:

\( \frac{F_e}{\rho_B V g} = \frac{F_e}{(\rho_B - \rho_L) V g} \).

Simplifying:

\( \rho_B V g = (\rho_B - \rho_L) V g \).

Cancelling \( V g \) from both sides:

\( \rho_B = \rho_B - \rho_L \Rightarrow \rho_B - \rho_L = k \cdot \rho_B \).

Substituting the given values:

\( \rho_B = 1.4 \, g/cm^3 \), \( \rho_L = 0.7 \, g/cm^3 \).

From the ratio:

\( k = \frac{\rho_B}{\rho_L} = \frac{1.4}{0.7} = 2 \).

Final Answer: The dielectric constant of the liquid is: \( k = 2 \).