Question

Three concentric metallic spherical shells of radii $R, 2R$ and $3R$ are given charges $Q_1, Q_2, Q_3$ respectively. It is found that the surface charge densities on the outer surface of the 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 : 4 : 9
• C. 1 : 2 : 3
• D. 1 : 3 : 5

Answer 1

Answer: (D)

Since surface charge densities on the outer surface of shells are same.

$\therefore \:\:\sigma = \frac{Q_{3}+Q_{2}+Q_{1}}{4\pi\left(3R\right)^{2}} = \frac{Q_{2}+Q_{1}}{4\pi\left(2R\right)^{2}}=\frac{Q_{1}}{4\pi R^{2}}$
or $\frac{Q_{3} +Q_{2}+Q_{1}}{9}=\frac{Q_{2}+Q1}{4}$ ....(i)
$ \frac{Q_{2}+Q_{1}}{4} =\frac{Q_{1}}{1}$ ..(ii)
$ Q_{1} +Q_{2} =4Q_{1}$ or $ Q_{2} =3 Q_{1}$ ....(iii)
Also $\frac{Q_{3}+Q_{2}+Q_{1}}{9} =\frac{Q_{1}}{1} $
$ Q_{3} +4Q_{1} =9Q_{1} $ or $Q_{3} = 5Q_{1}$ ...(iv)
$ \therefore \:\:\:\:\:Q_{1} : Q_{2} : Q_{3} = 1 : 3 : 5$