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Three concentric metallic spherical shells of radii $R, 2R, 3R$, are given charges $Q_1, Q_2, Q_3$, respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, $Q_1 : Q_2 : Q_3$ is
$1 : 2 : 3$
$1 : 3 : 5$
$1 : 4 : 9$
$1 : 8 : 18$
Solution
Due to induction net charges on outer surface of spheres are as shown.
$\sigma =\frac{\mathrm{Q}_{1}}{4 \pi \mathrm{R}^{2}}=\frac{\mathrm{Q}_{1}+\mathrm{Q}_{2}}{4 \pi(2 \mathrm{R})^{2}}=\frac{\mathrm{Q}_{1}+\mathrm{Q}_{2}+\mathrm{Q}_{3}}{4 \pi(3 \mathrm{R})^{2}}$
$\Rightarrow \mathrm{Q}_{1}=\frac{\mathrm{Q}_{1}+\mathrm{Q}_{2}}{4}=\frac{\mathrm{Q}_{1}+\mathrm{Q}_{2}+\mathrm{Q}_{3}}{4 \pi(3 \mathrm{R})^{2}}$
$\Rightarrow \mathrm{Q}_{2}=3 \mathrm{Q}_{1}$ and $\mathrm{Q}_{3}=5 \mathrm{Q}_{1}$
$\therefore \mathrm{Q}_{1}: \mathrm{Q}_{2}: \mathrm{Q}_{3}=1: 3: 5$
