Question

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

• A. $1:2:3$
• B. $1:3:5$
• C. $1:4:9$
• D. $1:8:18$

Answer 1

The correct option is B $1:3:5$

The charge distribution is shown in the figure below.


We know that the surface charge density on the outer surface of all the spheres is same.
The surface charge density on the outer sphere of radius $R$ is
$\sigma =\frac{Q_1}{4\pi R^2}$ -----------(1)
The surface charge density on the outer sphere of radius $2R$ is
$\sigma =\frac{Q_2+Q_1}{16\pi R^2}$ ----------(2)
The surface charge density on the outer sphere of radius $3R$ is
$\sigma =\frac{Q_2+Q_1+Q_3}{36\pi R^2}$ ----------(3)
From 1, 2 $1=\frac{4Q_1}{Q_2+Q_1}\Rightarrow \frac{Q_2+Q_1}{4Q_1}=1$
$\frac{Q_2}{Q_1}=3$ -----------(4)

from 2, 3
$1=\frac{9(Q_2+Q_1)}{4(Q_2+Q_1+Q_3)}$
from eq. (1) $Q_2=3Q_1$
then $\frac{Q_3}{Q_1}=5$----------(5)
from (4) and (5)
$Q_1:Q_2:Q_3=1:3:5$