Question

The distance between the plates of a circular parallel plate capacitor (PPC) of diameter $40\text{mm}$, whose capacity is equal to that of an isolated metallic sphere of radius $1\text{m}$ will be

• A. $0.01\text{mm}$
• B. $0.1\text{mm}$
• C. $1\text{mm}$
• D. $10\text{mm}$

Answer 1

The correct option is B $0.1\text{mm}$

Given,
Radius of the isolated spherical capacitor, $R=1\text{m}$

Radius of circular plates of PPC, $r=\frac{40}{2}=20\text{mm}$

Capacitance of isolated spherical capacitor is given by,
$$C_{sphere}=4\pi {\epsilon }_{0}R=4\pi {\epsilon }_0\times 1...\left(1\right)$$Capacitance of parallel plate capacitor is given by, $$C_{ppc}=\frac{{\epsilon }_{0}A}{d}=\frac{{\epsilon }_0\pi {(20\times {10}^{-3})}^2}{d}...\left(2\right)$$ According to the problem statement,

$C_{sphere}=C_{ppc}$

$\Rightarrow 4\pi {\epsilon }_0\times 1=\frac{{\epsilon }_0\pi {(20\times {10}^{-3})}^2}{d}$

$\Rightarrow 4\pi {\epsilon }_0\times 1=\frac{4\pi {\epsilon }_0\times {10}^{-4}}{d}$

$\Rightarrow d={10}^{-4}\text{m}=0.1\text{mm}$

Hence, option (b) is the correct answer.

Key Concept- Capacitance of parallel plate capacitor and capacitance of an isolated metallic sphere.