JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Capacitance

Capacitance of an isolated conducting sphere of radius

R_{1}

becomes

n

times when it is enclosed by a concentric conducting sphere of radius

R_{2}

connected to earth. The ratio of their radii

(\frac{R _{2}}{R _{1}})

is:

Answer: (A)

Solution

The capacitance of an isolated conducting sphere of radius

R_{1}

is:

C_{1} = 4 π ε_{0} R_{1}

When enclosed by a concentric sphere of radius

R_{2}

connected to earth, the capacitance becomes:

C_{2} = \frac{4 π ε _{0} R _{1} R _{2}}{R _{2} - R _{1}}

Given

C_{2} = n C_{1}

\frac{4 π ε _{0} R _{1} R _{2}}{R _{2} - R _{1}} = n \times 4 π ε_{0} R_{1}

Simplifying:

\frac{R _{2}}{R _{2} - R _{1}} = n ⟹ R_{2} = n (R_{2} - R_{1}) ⟹ R_{2} = n R_{2} - n R_{1}

n R_{1} = (n - 1) R_{2} ⟹ \frac{R _{2}}{R _{1}} = \frac{n}{n - 1}

JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Single correct option