A parallel plate air capacitor consists of two circular plates of diameter 4 cm. At what distance should the plates be held so as to have the same capacitance as that of a sphere of diameter 16 cm?

1.25 mm

The correct answer is Option (4) → 1.25 mm

Capacitance of a sphere of radius $R$:

$C_\text{sphere} = 4 \pi \epsilon_0 R$

Diameter of sphere = 16 cm ⇒ radius $R = 0.08 \, \text{m}$

$C_\text{sphere} = 4 \pi \epsilon_0 \cdot 0.08 = 0.32 \pi \epsilon_0$

Capacitance of a parallel plate capacitor:

$C_\text{plate} = \frac{\epsilon_0 A}{d}$

Plate diameter = 4 cm ⇒ radius $r = 0.02 \, \text{m}$

$A = \pi r^2 = \pi (0.02)^2 = 4 \times 10^{-4} \pi \, \text{m}^2$

Equate $C_\text{plate} = C_\text{sphere}$:

$\frac{\epsilon_0 A}{d} = 0.32 \pi \epsilon_0$

$d = \frac{A}{0.32 \pi} = \frac{4 \times 10^{-4} \pi}{0.32 \pi} = 1.25 \times 10^{-3} \, \text{m}$

d $\approx 1.25 \, \text{mm}$