Two non-conducting solid spheres of radii $R$ and $2 R$ , having uniform volume charge densities $ρ_{1}$ and $ρ_{2}$ respectively, touch each other. The net electric field at a distance $2 R$ from the centre of the smaller sphere, along the line joining the centre of the sphere is zero. The ratio $\frac{ρ_{1}}{ρ_{2}}$ can be

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$- \frac{32}{25}$

$- 4$

$\frac{32}{25}$

$4$

answer is B, D.

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Detailed Solution

At point $P$

If resultant electric field at M is zero, then

$\frac{K Q_{1}}{4 R^{2}} = \frac{K Q_{2}}{8 R^{3}} R \Rightarrow \frac{ρ_{1}}{ρ_{2}} = 4$

At point $Q$

If resultant electric field is zero, then

$\frac{K Q_{1}}{4 R^{2}} + \frac{K Q_{2}}{25 R^{2}} = 0$

$\Rightarrow \frac{ρ_{1}}{ρ_{2}} = - \frac{32}{25}$ ( $ρ_{1}$ must be negative)

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