Two non-conducting spheres of radii $R_{1}$ and $R_{2}$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,

(C) the electrostatic field is constant in magnitude.; (D) the electrostatic field has same direction.

(A) the electrostatic field is zero.

(B) the electrostatic potential is constant.

JEE Advanced · 2013 · 02 Jun 2013 · Paper 2 · Electrostatics

Two non-conducting spheres of radii

R_{1}

and

R_{2}

and carrying uniform volume charge densities

+ ρ

and

- ρ

, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,

Answer: (C), (D)

For a uniformly charged sphere, the electric field at an internal point P at distance

r

from center is

E = \frac{ρ r}{3 ε _{0}}

.

In the overlapping region, a point P experiences:

Field from sphere 1: $E_{1} = \frac{ρ x}{3 ε _{0}}$ (where $x$ is position from center 1)
Field from sphere 2: $E_{2} = \frac{- ρ y}{3 ε _{0}}$ (where $y$ is position from center 2)

Net field:

E = E_{1} + E_{2} = \frac{ρ ( x - y )}{3 ε _{0}} = \frac{ρ d}{3 ε _{0}}

where

d

is the vector from center 2 to center 1 (constant). Thus the field is uniform in magnitude and direction throughout the overlapping region.

JEE Advanced · 2013 · 02 Jun 2013 · Paper 2 · One or more correct options