Question

The capacity of an isolated spherical conductor of radius $a$ is

• A. $\frac{a}{4\pi {ϵ}_0}Farad$
• B. $4\pi {ϵ}_{0}aFarad$
• C. $\frac{4\pi {ϵ}_0}{a}Farad$
• D. $aFarad$

Answer 1

The correct option is B $4\pi {ϵ}_{0}aFarad$

Capacitance of a spherical capacitor $C=\frac{4\pi {ϵ}_o{R}_1{R}_2}{R_2-R_1}$
For an isolated capacitor, $R_2=\infty$ and $R_1=a$ (given)
Thus $R_1$ can be neglected compared to $R_2$.
$⟹$ $C=\frac{4\pi {ϵ}_o{R}_1{R}_2}{R_2}=4\pi {ϵ}_o{R}_1$
Thus capacitance $C=$$4\pi {ϵ}_{o}a$ $Farad$