Two identical non-conducting solid spheres of same mass and charge are suspended in the air from a common point by two non-conducting, massless strings of the same length. At equilibrium, the angle between the strings is . The spheres are now immersed in a dielectric liquid of density and dielectric constant . If the angle between the strings remains the same after the immersion, then
electric force between the spheres remains unchanged
the electric force between the spheres reduces
the mass density of the spheres is
the tension in the strings holding the spheres remains unchanged
the mass density of the spheres is
Step 1: Given data
The angle between the strings is
The density of the liquid,
Dielectric constant, .
Here, is the mass of non-conducting solid spheres, is the acceleration due to gravity, is the mass density of the spheres, is the mass of the spheres, and is the relative permittivity of the medium.
Step 2: In the case of option B,
The force between two charges is given by Coulomb's law as,
Where and are the two charges, is the distance between them, is the permittivity of free space and is the relative permittivity (also called the dielectric constant) of the medium.
The dielectric constant of air is .
So the electric force between the charges in air is,
When the system is immersed in the liquid whose dielectric constant is , the electric force between the charges is
Thus,
Hence, option B is correct.
Step 3: In case of option C
From Newton's second law of motion, we have that,
Where is the mass of the object and is the acceleration of the object when force is applied on the object.
At equilibrium, the net force along the y-axis is
where is the tension in the string
When a liquid medium is introduced, the buoyant force acts opposite to gravity. So the above equation becomes,
Where is the buoyant force.
The buoyant force is given as,
Where is the density of the liquid, is the volume of the object the force is acting on.
Therefore,
Dividing by ,
At equilibrium, the net force along the x-axis is
Thus in air,
In the liquid medium,
But,
So,
Substituting in , we have,
Hence, option C is also correct.
Therefore, only options B and C are correct.