Question

Three concentric metallic spherical shells of radii $R, \, 2R, \, 3R$ are given $Q_{1}, \, Q_{2}, \, Q_{3}$ respectively. It is found that the surface charge densities on the outer surfaces of shells are equal. Then, the ratio of the charges given to the shells $Q_{1}:Q_{2}:Q_{3}$ is

• A. $1:8:18$
• B. $1:3:5$
• C. $3:4:5$
• D. $1:4:9$

Answer 1

Answer: (B)

Due to induction, net charge on outer surfaces of spheres will be as
$\sigma =\frac{Q_{1}}{4 \pi R_{}^{2}}=\frac{Q_{1} + Q_{2}}{4 \pi \left(2 R\right)^{2}}=\frac{Q_{1} + Q_{2} + Q_{3}}{4 \pi \left(3 R\right)^{2}}$
$\Rightarrow Q_{1}=\frac{Q_{1} + Q_{2}}{4}=\frac{Q_{1} + Q_{2} + Q_{3}}{9}$
On solving, we get
$Q_{2}=3Q_{1}$ and $Q_{3}=5Q_{1}$
Therefore, $Q_{1}:Q_{2}:Q_{3}=1:3:5$