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  $\frac{{\rho }_1}{{\rho }_2}$ can be 

Electric field  $E_1$ due to smaller sphere ‘A’ at Q $E_1=\frac{1}{4\pi {\epsilon }_0}\frac{\frac{4}{3}\pi R^3{\rho }_1}{{\left(2R\right)}^2}$ 
Electric field  $E_2$  due to bigger sphere’B’ at Q 
   

$$\begin{gathered} E_2=\frac{1}{4\pi {\epsilon }_0}\frac{\frac{4}{3}\pi {\left(2R\right)}^3{\rho }_2}{{\left(5R\right)}^2} \\ {E}_1+E_2=0 \frac{1}{4\pi {\epsilon }_0}\frac{\frac{4}{3}\pi R^3{\rho }_1}{{\left(2R\right)}^2}+\frac{1}{4\pi {\epsilon }_0}\frac{\frac{4}{3}\pi {\left(2R\right)}^3{\rho }_2}{{\left(5R\right)}^2}=0 \\ \therefore \frac{{\rho }_1}{{\rho }_2}=-\frac{32}{25} \end{gathered}$$

Electric field  $E_1$  due to smaller sphere ‘A’ at P
 $E_1=\frac{1}{4\pi {\epsilon }_0}\frac{{\rho }_1\times \frac{4}{3}\pi R^3}{{\left(2R\right)}^2}=\frac{1}{4\pi {\in }_0}\times \frac{{\rho }_1\pi R}{3}=\frac{{\rho }_{1}R}{4{\in }_0\times 3}$
Electric field  ${\text{E}}_{\text{2}}\text{dueto}$  bigger sphere ‘B’ at P
 $$\begin{gathered} {\text{E}}_{\text{2}}=\frac{1}{4\pi {\epsilon }_0}\frac{\frac{4}{3}\pi {\left(2R\right)}^3{\rho }_{2}R}{{\left(2R\right)}^3}=\frac{{\rho }_{2}R}{3{\epsilon }_0} \\ {E}_1=E_2\therefore \frac{{\rho }_{1}R}{4{\in }_0\times 3}=\frac{{\rho }_{2}R}{3{\in }_0}\Rightarrow \frac{{\rho }_1}{{\rho }_2}=4 \end{gathered}$$