Ncert Exercises & Solutions

A spherical capacitor has an inner sphere of radius \(12~\text{cm}\) and an outer sphere of radius \(13~\text{cm}.\) The outer sphere is earthed, and the space between the concentric spheres is filled with a liquid of dielectric constant \(32.\) What is the capacitance of this capacitor, the potential of the inner sphere if it carries a charge of \(2.5~\mu\text C,\) and how does its capacitance compare to that of an isolated sphere of radius \(12~\text{cm}?\)

1.	\(C\approx 5.5\times 10^{-9}~\text F,~V_{inner} \approx 4.5 \times 10^2,~\text{Capacitance is larger by a factor of} \sim 400.\)
2.	\(C\approx 5.5\times 10^{-9}~\text F,~V_{inner} \approx 4.5 \times 10^2,~\text{Capacitance is larger by a factor of} \sim 40.\)
3.	\(C\approx 5.5\times 10^{-9}~\text F,~V_{inner} \approx 4.5 \times 10^2,~\text{Capacitance is larger by a factor of} \sim 41.\)
4.	\(C\approx 2.75\times 10^{-9}~\text F,~V_{inner} \approx 9.1 \times 10^2,~\text{Capacitance is larger by a factor of} \sim 20.\)

Radius of the inner sphere, r₂ = 12 cm = 0.12 m

Radius of the outer sphere, r₁ = 13 cm = 0.13 m

Charge on the inner sphere, q = 2.5 UC = 2.5 x 10^-6 C

The dielectric constant of a liquid,

\in_{r} = 32

(a) Capacitance of the capacitor is given by the relation,

C = \frac{4 {πϵ}_{0} ϵ_{r} r_{1} r_{2}}{r_{1} - r_{2}}

Where,

ε_{0}

= Permittivity of free space = 8.85 x 10^-12 C² N^-1 m^-2

\begin{array}{l} \frac{1}{4 {πϵ}_{0}} = 9 \times 10^{\circ} {Nm}^{2} C^{- 2} \\ ∴ C = \frac{32 \times 0 .12 \times 0 .13}{9 \times 10^{9} \times (0 .13 - 0 .12)} \\ \approx 5 .5 \times 10^{- 9} F \end{array}

Hence, the capacitance of the capacitor is approximately 5.5 X 10^-9 F.

(b) Potential of the inner sphere is given by,

\begin{aligned} V & = \frac{q}{C} \\ = \frac{2 .5 \times 10^{- 6}}{5 .5 \times 10^{- 9}} = 4 .5 \times 10^{2} V \end{aligned}

Hence, the potential of the inner sphere is 4.5 X 10² V.

The capacitance of the sphere is given by the relation,

\begin{aligned} C^{'} & = 4 {πϵ}_{0} r \\ = 4 π \times 8 .85 \times 10^{- 12} \times 12 \times 10^{- 12} \\ = 1 .33 \times 10^{- 11} F \end{aligned}

The capacitance of the isolated sphere is less in comparison to the concentric spheres. This is because the outer sphere of the concentric spheres is earthed. Hence, the potential difference is less and the capacitance is more than the isolated sphere.

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