A solid sphere of radius R₁ and volume charge density $ρ = \frac{ρ_{0}}{r}$ is enclosed by a hollow sphere of radius R₂ with negative surface charge density $σ$ , such that the total charge in the system is zero. P₀ is positive constant and r is the distance from the center of the sphere. The ratio $R_{2} / R_{1}$ is

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$\sqrt{\frac{ρ_{0}}{3 σ}}$

$\sqrt{\frac{ρ_{0}}{σ}}$

$\sqrt{\frac{2 ρ_{0}}{3 σ}}$

$\sqrt{\frac{ρ_{0}}{2 σ}}$

answer is D.

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Detailed Solution

Charge inside sphere of radius $R_{1}$ $Q_{1} = \int ρ \cdot 4 {πr}^{2} dr = \int_{0}^{R_{1}} \frac{ρ_{0}}{r} 4 {πr}^{2} dr = 4 {πρ}_{0} \int_{0}^{R_{1}} rdr$
$= \frac{4 {πρ}_{0} R_{1}^{2}}{2} = 2 {πρ}_{0} R_{1}^{2}$

Charge on hollow sphere $Q_{2} = - σ . 4 {πR}_{2}^{2}$

$As we know$ $Q_{1} + Q_{2} = 0$

$2 {πρ}_{0} R_{1}^{2} - σ . 4 {πR}_{2}^{2} = 0 \Rightarrow \frac{R_{2}}{R_{1}} = \sqrt{\frac{ρ_{0}}{2 σ}}$

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