By Sunil Bhardwaj

1069 Views

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.

#### Latest News

• Become an Instructor 4 March, 2018

Apply to join the passionate instructors who share their expertise and knowledge with the world. You'll collaborate with some of the industry's best producers, directors, and editors so that your content is presented in the best possible light..

#### More Chapters

• Electrostatics
• Optics
• Electricity and Magnetism
• #### Other Subjects

• English
• Applied Physics
• Environmental Studies
• Physical Chemistry
• Analytical Chemistry
• Organic Chemistry
• Soft Skills
• Engineering Drawing
• General Medicine
• Mathematics
• Patente B Italia