Question

The capacitance of a spherical conductor with radius 1 m is :
A. $9 \times {10^9}F$
B. $1\mu f$
C. $1.1 \times {10^{ - 10}}F$
D. $1 \times {10^{ - 8}}F$

Answer 1

Hint: Spherical capacitors are spherical in shape therefore capacitance of a spherical conductor is given by the formula,
$C = 4\pi { \in _{_0}}R$
Where ${ \in _0} = $ permittivity of free space
               R = Radius of conductor
               C = Capacitance of conductor
Capacitance of a spherical conductor can easily be calculated by finding the potential of the spherical conductor.

Complete step-by-step answer:
Given: Radius of capacitor = 1$m$
We know that,
Electric field due to a spherical capacitance is given by,
$E = \dfrac{Q}{{4\pi { \in _0}{R^2}}}$
We know that,
$V = E \times R \Leftrightarrow V = \dfrac{Q}{{4\pi { \in _0}{R^2}}} \times R$
$\therefore V = \dfrac{Q}{{4\pi { \in _0}R}} \cdot \cdot \cdot \cdot \cdot \cdot \left( 1 \right)$
Also we know that,
$C = \dfrac{Q}{V} \cdot \cdot \cdot \cdot \cdot \cdot \left( 2 \right)$
From (1) and (2) we get
$C = \dfrac{Q}{{\dfrac{Q}{{4\pi { \in _0}R}}}} \Leftrightarrow C = 4\pi { \in _0}R$
Hence spherical capacitance of a spherical conductor given by
$C = 4\pi { \in _0}r$
But $\dfrac{1}{{4\pi { \in _0}}} = 9 \times {10^9}$
$\therefore 4\pi { \in _0} = 9 \times {10^{ - 9}}$$\therefore 4\pi { \in _0} = \dfrac{1}{9} \times {10^{ - 9}}$
$\therefore $Required capacitance of conductor
$C = \dfrac{1}{9} \times {10^{ - 9}} \times (1)$
On solving we get
$C = 1.1 \times {10^{ - 10}}F$
Hence the spherical capacitance of a conductor of radius 1$m$is $1.1 \times {10^{ - 10}}$

Note: A spherical capacitor consist of a hollow or a solid spherical conductor surrounded by another concentric hollow spherical conductor is given by
$C = \dfrac{{4\pi { \in _0}{r_1}{r_{{2_{}}}}}}{{{r_1} - {r_2}}}$
where ${r_1}$= outer radius of conductor
             ${r_{_2}}$= inner radius of conductor
Relative permeability is used for capacitor having dielectric constant k, given by
${ \in _r} = { \in _0}k$.