JEE Advanced · 2013 · 02 Jun 2013 · Paper 1 · Electrostatics

There are two possible positions where the net field is zero: point P (outside both spheres) and point P' (inside the larger sphere).

For point P (at distance 2R from center of smaller sphere, outside both):

Field due to smaller sphere: $E_1=\frac{k{\rho }_1\frac{4}{3}\pi R^3}{(2R{)}^2}$

Field due to larger sphere (at distance R from its center): $E_2=\frac{{\rho }_{2}R}{3{ϵ}_0}$

For $E_1=E_2$:

$\frac{{\rho }_1{R}^3}{4R^2\times 3{ϵ}_0}=\frac{{\rho }_{2}R}{3{ϵ}_0}$

$\frac{{\rho }_1}{{\rho }_2}=4$ ✓ (D is correct)

For point P' (inside the larger sphere):

At distance 5R from center of larger sphere:

$\frac{k{\rho }_1\frac{4}{3}\pi R^3}{(4R{)}^2}=\frac{k{\rho }_2\frac{4}{3}\pi (5R{)}^3}{(5R{)}^2}$

Simplifying:

$\frac{{\rho }_1}{{\rho }_2}=-\frac{32}{25}$ ✓ (B is correct)

Hence the correct answers are (B) and (D).