Given:
- Two identical spherical charged metallic spheres
- Hung by strings
- Strings make 60 degree with each other in vacuum
- When the arrangement is kept in a liquid of density 0.5 gram per CC, the angle remains the same
- Density of a sphere is equal to 1.5 gram per cc

To find:
- Dielectric constant of the fluid

Solution:

1. Electric field due to a charged sphere:
- Electric field at a point outside a charged sphere is given by:
E = kq/r^2
- Where k is Coulomb's constant, q is the charge on the sphere and r is the distance between the point and the center of the sphere
- Electric field at a point inside a charged sphere is given by:
E = kq*r/R^3
- Where R is the radius of the sphere and r is the distance between the point and the center of the sphere
- Electric field is a vector quantity and its direction is radially outward from the center of the sphere

2. Electric potential due to a charged sphere:
- Electric potential at a point outside a charged sphere is given by:
V = kq/r
- Where k is Coulomb's constant, q is the charge on the sphere and r is the distance between the point and the center of the sphere
- Electric potential at a point inside a charged sphere is given by:
V = kq*(3R^2 - r^2)/(2R^3)
- Where R is the radius of the sphere and r is the distance between the point and the center of the sphere

3. Equilibrium of charged spheres:
- When two charged spheres are in equilibrium, the net electrostatic force on each sphere is zero
- This means that the electric field at any point on the surface of each sphere is perpendicular to the surface of the sphere
- The angle between the strings holding the spheres is equal to the angle between the lines joining the centers of the spheres to the point of intersection of the strings

4. Effect of a medium on electric field and potential:
- When a charged sphere is placed in a medium, the electric field and potential around the sphere are modified due to the presence of the medium
- The modification is characterized by the dielectric constant of the medium, which is a measure of the ability of the medium to store electric charge

5. Calculation of dielectric constant:
- Let the radius of each sphere be R and the charge on each sphere be q
- In vacuum, the electric field at the point of intersection of the strings is given by:
E_vacuum = kq/R^2
- Let the tension in each string be T
- In equilibrium, the horizontal component of the tension in each string is equal to the vertical component of the tension in the other string
- This gives:
T*sin(60) = T*cos(60)*tan(60)
- Solving for T, we get:
T = kq/R^2
- When the whole arrangement is kept in a liquid of density 0.5 gram per CC, the tension in each string remains the same
- This means that the magnitude of the force on each sphere due to the tension in the string remains the same
- Let the electric field at the point of intersection of the strings in the liquid be E_liquid and the dielectric constant of the liquid be ε
- The force