Question

Two charged balls of the same radius and weight suspended on threads of equal length are immersed into a liquid having density of $d_1$ and a dielectric constant $K$. The density $d$ of the material of the balls for the angles of divergence of the threads in the air and in the dielectric to be the same is:

• A. $\frac{Kd}{K-1}$
• B. $\frac{K-1}{Kd}$
• C. $\frac{d}{K-1}$
• D. $\frac{K-1}{d}$

Answer 1

The correct option is A $\frac{Kd}{K-1}$

When in air , for each ball ,

$Tcos\theta =Density\times volume\times g=dVg$ ( Vertical component of tension is balanced by weight )

$Tsin\theta =\frac{q^2}{4\pi {ϵ}_0(2lsin\theta {)}^2}$ (

The horizontal component of tension is balanced by the Electrostatic Force )

$tan\theta =\frac{\frac{q^2}{4\pi {ϵ}_0(2lsin\theta {)}^2}}{dVg}$ ----------- (1)
Similarly ,when immersed in liquid ,

$tan\theta =\frac{\frac{q^2}{4\pi K{ϵ}_0(2lsin\theta {)}^2}}{Vg(d-d_1)}$ ----------(2) ( Buoyant force, gravity and electrostatic force )
From (1) and (2)
$K(d-d_1)=d$

Hence , $d=\frac{K{d}_1}{K-1}$