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 centre of the spheres, is zero. The ratio ${\rho }_1/{\rho }_2$ can be

• A. $-4$
• B. $\frac{-32}{25}$
• C. $\frac{32}{25}$
• D. $4$

Answer 1

Answer: (B), (D)

At point $P_1,\frac{1}{4\pi {\epsilon }_0}\frac{{\rho }_1(4/3)\pi R^3}{4R^2}=\frac{{\rho }_{2}R}{3{\epsilon }_0}$
$\frac{{\rho }_{1}R}{12}=\frac{{\rho }_{2}R}{3}$
$\frac{{\rho }_1}{{\rho }_2}=4$
At point $P_2$
$\frac{{\rho }_1(4/3)\pi R^3}{(2R{)}^2}+\frac{{\rho }_2(4/3)\pi 8R^3}{(5R{)}^2}=0$
$\therefore \frac{{\rho }_1}{{\rho }_2}=-\frac{32}{25}$