Question

Establish the formula for the capacity of the spherical capacitor.

Answer 1

Step 1: Given:

A and B are two co-centric hollow spheres separated by a dielectric medium between them

B is connected to earth $+q$charge is given to A and therefore $-q$charge is induced on B on its inner surface. The $q$charge on outer surface of B goes to earth.

Radii of A and B be $a$ and $b$.

Step 2: Calculate the capacity of the spherical capacitor:

The potential of A is $V_A=\text{potential due to charge on}A+\text{Potential due to charge on}B$.
$$\begin{gathered} V_a=\frac{q}{4\pi {\epsilon }_{0}ka}-\frac{q}{4\pi {\epsilon }_{0}kb} \\ =\frac{q}{4\pi {\epsilon }_{0}k}\left(\frac{1}{a}-\frac{1}{b}\right) \\ =\frac{q}{4\pi {\epsilon }_{0}k}\left(\frac{b-a}{ab}\right) \end{gathered}$$
We know
$$\begin{gathered} C=\frac{q}{V_a} \\ =\frac{q}{q}\times \frac{ab\times 4\pi {\epsilon }_0}{b-a} \\ =4\pi {\epsilon }_{0}k\left[\frac{ab}{b-a}\right] \end{gathered}$$
Therefore, the capacity of the spherical capacitor will be, $C=4\pi {\epsilon }_{0}k\left[\frac{ab}{b-a}\right]$