# Derive an expression to calculate capacitance of spherical capacitor formed by two concentric shell of radii (a) and (b)?

Sunil Bhardwaj

CAPACITANCE OF SPHERICAL CAPACITOR

Consider a spherical capacitor consists of two spherical concentric shells. The inner spherical shell has radius "$$a$$" and charge $$+Q$$. The outer spherical shell has radius "$$b$$" and charge $$-Q$$ as shown in fig.

The electric field lines are radial and directed outward from inner shell (+ve plate) to outer shell (-ve plate).

Imagine a Gaussian sphere of radius $$r$$ which encloses inner spherical amount of flux diverging out of Gaussian sphere through small area element $$da$$ of this Gaussian surface. $$d\phi_c=\ \vec{E}.\vec{da}\$$ $$d\phi_c=\ E.da\ \cos{0}\$$ $$d\phi_c=\ E.da\$$ Net flux through whole Gaussian sphere is $$\phi_c\ =\oint{E.da}$$ $$\phi_c=\ E.\left(4\pi r^2\right)\$$

. . . . . . . . . . . . . (1)

Gauss's law is $$\phi_c\ =\ \frac{Q}{\varepsilon_0}$$

. . . . . . . . . . . . . (2)

Comparing eq(1) and eq(2) $$E.\left(4\pi r^2\right)\ =\ \frac{Q}{\varepsilon_0}$$ $$E\ =\ \frac{Q}{\left(4\pi r^2\right)\varepsilon_0}$$ It is the magnitude of electric field at any point on the Gaussian sphere due to uniform charge distribution of spherical capacitor.

Potential difference between the shells of the capacitor is $$V\ =\ \int_{a}^{b}{E\ ds}$$ $$V\ =\ \int_{a}^{b}{\frac{Q}{\left(4\pi r^2\right)\varepsilon_0}\ dr}$$ $$V\ =\ \frac{Q}{\left(4\pi\right)\varepsilon_0}\int_{a}^{b}{r^{-2}\ dr}$$ $$V\ =\ \frac{Q}{\left(4\pi\right)\varepsilon_0}\left[\frac{r^{-2+1}}{-2+1}\right]_a^b$$ $$V\ =\ \frac{Q}{\left(4\pi\right)\varepsilon_0}\left[\frac{r^{-1}}{-1}\right]_a^b$$ $$V\ =\ \frac{Q}{\left(4\pi\right)\varepsilon_0}\left[\frac{-1}{r}\right]_a^b$$ $$V\ =\ \frac{Q}{\left(4\pi\right)\varepsilon_0}\left[\frac{1}{a}-\frac{1}{b}\right]$$ $$V\ =\ \frac{Q}{\left(4\pi\right)\varepsilon_0}\left[\frac{b-a}{ab}\right]$$ $$\frac{Q}{V}=\ \left(4\pi\right)\varepsilon_0\left[\frac{ab}{b-a}\right]$$

. . . . . . . . . . . . . (3)

The capacitance of spherical capacitor is $$\frac{Q}{V}=C$$

. . . . . . . . . . . . . (4)

Comparing eq (3) and eq (4) $$C=\ \left(4\pi\right)\varepsilon_0\left[\frac{ab}{b-a}\right]$$ This is capacitance of spherical capacitor. It depends upon radius of inner shell and radius of outer spherical shell of capacitor.

ISOLATED SPHERE:

The positively charged inner spherical shell is called isolated sphere negatively charged outer spherical shell is moved at infinity distance. The potential is $$V\ =\ \frac{Q}{\left(4\pi\right)\varepsilon_0}\left[\frac{1}{a}-\frac{1}{b}\right]$$ $$V\ =\ \frac{Q}{\left(4\pi\right)\varepsilon_0}\left[\frac{1}{a}-\frac{1}{\infty}\right]$$ $$V\ =\ \frac{Q}{\left(4\pi\right)\varepsilon_0a}$$ $$\frac{Q}{V}=\ \left(4\pi\right)\varepsilon_0a$$ $$C=\ \left(4\pi\right)\varepsilon_0a$$ This is capacitance of isolated sphere.