Q. 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
NTA AbhyasNTA Abhyas 2022
Solution:
Field due to small sphere is given as $E_{1}=\frac{K \left(\rho \right)_{1} \frac{4}{3} \pi R^{3}}{\left(2 R\right)^{2}}$ , where $K$ is a constant.
Field due to bigger sphere is given as $E_{2}=\frac{K \left(\rho \right)_{2} \frac{4}{3} \pi \left(2 R\right)^{3} \times R}{\left(2 R\right)^{3}}$
Equating above two, we get $\frac{K \left(\rho \right)_{1} \frac{4}{3} \pi R^{3}}{\left(2 R\right)^{2}}=\frac{K \left(\rho \right)_{2} \frac{4}{3} \pi \left(2 R\right)^{3} \times R}{\left(2 R\right)^{3}}\Rightarrow \frac{\left(\rho \right)_{1}}{\left(\rho \right)_{2}}=4$
Hence, the ratio of volume charge densities is $4$
Field due to bigger sphere is given as $E_{2}=\frac{K \left(\rho \right)_{2} \frac{4}{3} \pi \left(2 R\right)^{3} \times R}{\left(2 R\right)^{3}}$
Equating above two, we get $\frac{K \left(\rho \right)_{1} \frac{4}{3} \pi R^{3}}{\left(2 R\right)^{2}}=\frac{K \left(\rho \right)_{2} \frac{4}{3} \pi \left(2 R\right)^{3} \times R}{\left(2 R\right)^{3}}\Rightarrow \frac{\left(\rho \right)_{1}}{\left(\rho \right)_{2}}=4$
Hence, the ratio of volume charge densities is $4$