Question

Figure shows three concentric conducting spherical shells with inner and outer shells earthed and the middle shell is given a charge $q$ . Let, $q_1$ be the charge accumulated on sphere 1 and $q_3$ be the charge accumulated on sphere 3, then match the column 1 with column 2 (where, $K$ is Coulomb's constant)

S.No	Column 1	S.No	Column 2
1.	$q_1$	a.	0
2.	$q_3$	b.	$\frac{-4}{25}q$
3.	Electrostatic energy of the system stored in the region II.	c.	$\frac{-21}{25}q$
		d.	$-q$
		e.	$\frac{2K{\left(q\right)}^2}{35r}$
		f.	$\frac{2K{(q_1+q)}^2}{35r}$

• A. $1\to c,2\to b,3\to e$
• B. $1\to b,2\to c,3\to e$
• C. $1\to b,2\to c,3\to f$
• D. $1\to d,2\to d,3\to a$

Answer 1

The correct option is C $1\to b,2\to c,3\to f$

Potential inside an isolated hollow sphere is constant. Concentric hollow sphere potential at a point can be added due to principle of superposition. Grounded Surface will have zero potential.

For sphere 1,
$\frac{kq}{r_2}+\frac{{kq}_1}{r_1}+\frac{{kq}_3}{r_3}=0$

$\frac{2q}{5r}+\frac{q_1}{r}+\frac{2q_3}{7r}=0$

$14q+35q_1+10q_3=0$ --(1)


For sphere 3,
$\frac{{kq}_1}{r_3}+\frac{kq}{r_3}+\frac{{kq}_3}{r_3}=0$
$q_1+q+q_3=0$ --(2)

From (1) & (2),
$\Rightarrow q_1=\frac{-4}{25}q$
& $q_3=\frac{-21}{25}q$

$C_{II}=4\pi {ϵ}_0{\left[\frac{(3.5r-2.5r)}{3.5r\times 2.5r}\right]}^{-1}=\frac{35r}{4k}$

$U_{II}=\frac{{(q_1+q)}^2}{2C}=\frac{2k{(q_1+q)}^2}{35r}$