Question

The expression for capacity of a spherical conductor is.

• A. $C=\frac{1}{4\mu {ϵ}_{0}R}$
• B. $C=4\pi {ϵ}_{0}R$
• C. $C=4\pi {ϵ}_0{R}^2$
• D. $C=4\pi {ϵ}_0{R}^3$

Answer 1

Answer: (B)

$C=4\pi {ϵ}_{0}R$

To calculate capacity of a spherical conductor, first calculate capacity for spherical shell:
onsider a spherical shell with inner radius $a$ and out radius $b$. Now by applying Gauss' law to an charged conducting sphere, the electric field outside it is found to be :
$E=\frac{Q}{4\pi {ϵ}_0{r}^2}$
The voltage between the spheres can be found by integrating the electric field along a radial line:
$△V=\frac{Q}{4\pi {ϵ}_0}{\int }_a^b\frac{1}{r^2}dr=\frac{Q}{4\pi {ϵ}_0}[\frac{1}{a}-\frac{1}{b}]$
From the defination of capacity :
$C=\frac{Q}{△V}=\frac{4\pi {ϵ}_0}{[\frac{1}{a}-\frac{1}{b}]}$
Now for sphere capacitor of radius $R$, $a\to R$ and $b\to \infty$
$\therefore C=4\pi {ϵ}_{0}R$