Question

The capacitance of an isolated conducting sphere of radius R is proportional to

• A. $2R$
• B. $R$
• C. $R^2$
• D. $\frac{1}{R}$

Answer 1

The correct option is B

$R$

The explanation for the correct option:

Option B:

Step 1: Given

The radius of the sphere = $R$

Step 2: Determine the capacity of the given sphere

The capacity of an isolated conducting sphere of radius $R$ is given by,

$C=4{\mathrm{\pi ϵ}}_0\mathrm{R}$

(Where $4{\mathrm{\pi ϵ}}_0$ is the permittivity of free space)

Step 3: Determine the potential on the sphere

The potential on a sphere with radius R is given by

$V=\frac{1}{4{\mathrm{\pi ϵ}}_0}\frac{q}{R}$

(Where q is the charge)

Step 4: Substitute the value in the capacitance formula

The capacitance of a conductor is given by

$C=\frac{q}{V}$

(Where q is the charge and V is the potential)

$$\begin{gathered} C=\frac{\mathrm{q}\times 4{\mathrm{\pi ϵ}}_0\mathrm{R}}{\mathrm{q}} \\ =4{\mathrm{\pi ϵ}}_0\mathrm{R} \end{gathered}$$

We can observe from the formula that $C\propto R$.

Hence, option B is the correct option.