Question

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 $\frac{{\rho }_1}{{\rho }_2}$ is

• A. 2
• B. 8
• C. 4
• D. 5

Answer 1

Answer: (C)

Field due to small sphere is given as $E_1=\frac{K{\left(\rho \right)}_1\frac{4}{3}\pi R^3}{{\left(2R\right)}^2}$ , where $K$ is a constant.
Field due to bigger sphere is given as $E_2=\frac{K{\left(\rho \right)}_2\frac{4}{3}\pi {\left(2R\right)}^3\times R}{{\left(2R\right)}^3}$
Equating above two, we get $\frac{K{\left(\rho \right)}_1\frac{4}{3}\pi R^3}{{\left(2R\right)}^2}=\frac{K{\left(\rho \right)}_2\frac{4}{3}\pi {\left(2R\right)}^3\times R}{{\left(2R\right)}^3}\Rightarrow \frac{{\left(\rho \right)}_1}{{\left(\rho \right)}_2}=4$
Hence, the ratio of volume charge densities is $4$