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  4. Two non-conducting solid spheres of radii R and 2R , having uniform volume charge densities ρ 1 and ρ 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 ρ 1 / ρ 2 can be

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

JEE AdvancedJEE Advanced 2013Electric Charges and Fields

Solution:

At point $P_{1}, \frac{1}{4 \pi \varepsilon_{0}} \frac{\rho_{1}(4 / 3) \pi R^{3}}{4 R^{2}}=\frac{\rho_{2} R}{3 \varepsilon_{0}}$
$\frac{\rho_{1} R}{12}=\frac{\rho_{2} R}{3}$
$\frac{\rho_{1}}{\rho_{2}}=4$
At point $P _{2}$
$\frac{\rho_{1}(4 / 3) \pi R ^{3}}{(2 R )^{2}}+\frac{\rho_{2}(4 / 3) \pi 8 R ^{3}}{(5 R )^{2}}=0$
$\therefore \frac{\rho_{1}}{\rho_{2}}=-\frac{32}{25}$
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