Question

The capacitance of a spherical condenser is $1\mu F$. If the spacing between the two spheres is $1mm$, then the radius of the outer sphere is

• A. $30cm$
• B. $6m$
• C. $5cm$
• D. $3m$

Answer 1

The correct option is D

$3m$

Step 1: Given

Capacity, $$\begin{gathered} C=1\mu F \\ ={10}^{-6}F \end{gathered}$$

Spacing between two spheres, $$\begin{gathered} R_2-R_1=1mm \\ ={10}^{-3}m \end{gathered}$$

Step 2: Formula used

The capacity of a spherical capacitor bound by two spheres can be calculated by:

$C=\frac{4{\mathrm{\pi ϵ}}_0{\mathrm{R}}_1{\mathrm{R}}_2}{R_2-R_1}$

Where, $R_1,R_2$ are the radii of the spheres and ${ϵ}_0$ is the permittivity of the free space.

Step 3: Determine the radius

Since the spacing between the conductors is very small$\left(1mm\right)$, we can consider that both the capacitors have almost the same radii.

Therefore, $R_1\approx R_2$

Hence,

$$\begin{gathered} C=\frac{4{\mathrm{\pi ϵ}}_0{\mathrm{R}}_1{\mathrm{R}}_2}{R_2-R_1} \\ {10}^{-6}=\frac{4\mathrm{\pi }\times 9\times {10}^9\times {\mathrm{R}}_2\times {\mathrm{R}}_2}{{10}^{-3}} \\ {R}_2=3m \end{gathered}$$

Hence, option D is the correct option.