Two non-conducting solid spheres of radii R and 2R, having uniform volume charge densities `rho_1` and `rho_2` respectively, touch each other. The net electric field at a distance 2R from the centre of the smaller sphere, along the line joining the centres of the spheres, is zero. The ratio `rho_1/rho_2` can be

Options

A. `-4`
B. `-32/25`
C. `32/25`
D. 4

Answer

B, D

Explanation

First of all electric field intensity due to a non-conducting solid sphere of radius R with charge uniformly distributed over the volume can be written as follows:

x < R `rArr E = (rhox)/(3epsilon_0)` ..(i)
x > R `rArr E =(rho R^3)/(3epsilon_0x^2)`....(ii)
See the figure to understand the described question. There are two points on the line joining the centres of spheres, which are at a distance 2R apart from the centre of a smaller sphere. Assuming the polarities to be positive for both, we shall do a calculation and if they are of opposite polarity then the answer should indicate it. For example, let us proceed for the point P and we can see that both the fields are shown in the same direction but we require net field intensity to be zero. In the figure, `E_1` represents electric field intensity due to the smaller sphere and `E_2` represent the electric field intensity due to the bigger sphere.Point P : `E_1+E_2=0`
`rArr (rho_1R^3)/(3epsilon_0(2R)^2)+(rho_2(2R)^3)/(3epsilon_0(5R)^2)=0`
`rArr rho_1/rho_2=-((2)^3 (2)^2)/(5)^2 =-32/25`
Option (b) is correct
Point Q:
`E_1-E_2=0`
`rArr (rho_1R^3)/(3epsilon_0(2R)^2) -(rho_2(R))/(3epsilon_0) =0 rArr rho_1/rho_2=4`