The earth is assumed to be a charged conducting sphere having volume 'V' and surface area 'A'. The capacitance of the earth in free space is ( ϵ 0 = permittivity of free space)

Question

The earth is assumed to be a charged conducting sphere having volume 'V' and surface area 'A'. The capacitance of the earth in free space is ( ϵ 0 ​ = permittivity of free space)

TestHub · Accepted Answer

The capacitance of a spherical conductor of radius $R$ is given by $C = 4 π ϵ_{0} R$ .

The volume of the sphere is $V = \frac{4}{3} π R^{3}$ and its surface area is $A = 4 π R^{2}$ .

Taking the ratio of volume to surface area:

$\frac{V}{A} = \frac{\frac{4}{3} π R ^{3}}{4 π R ^{2}} = \frac{R}{3}$

This gives $R = \frac{3 V}{A}$ .

Substituting the value of $R$ in the expression for capacitance:

$C = 4 π ϵ_{0} (\frac{3 V}{A}) = 12 π ϵ_{0} \frac{V}{A}$

Answer: $12 π ϵ_{0} V / A$

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