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

Answer: (B)

$1:3:5$

In metallic shell , all charge will appear on the outer surface of outer shell and charge inside each is zero.
The charge distributions among the shells will occur in such a way that the charge inside each shell is zero and total charge will appear on the outer surface of outer shell. (shown in fig below)
Let surface density of charges of outer surfaces of each shell is $\sigma$.
now, $Q_1=\sigma .4\pi R^2$,
$Q_1+Q_2=\sigma .4\pi (2R{)}^2\Rightarrow Q_2=\sigma .4\pi 4R^2-Q_1=\sigma .4\pi 4R^2-\sigma .4\pi R^2$

$=3\left(\sigma .4\pi R^2\right)$

and
$Q_1+Q_2+Q_3=\sigma .4\pi (3R{)}^2$

$\Rightarrow Q_3=\sigma .4\pi 9R^2-(Q_1+Q_2)=\sigma .4\pi 9R^2-\sigma .4\pi 4R^2=5\left(\sigma .4\pi R^2\right)$
thus, $Q_1:Q_2:Q_3=1:3:5$