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?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 1.25 mm
Equate the capacitance of the parallel plate air capacitor, \( C = \varepsilon_0 A/d \), to the capacitance of an isolated sphere, \( C = 4\pi\varepsilon_0 R \).
\( \text{plate radius} = 4/2 = 2\,\mathrm{cm} \)
\( A = \pi (0.02)^{2}\,\mathrm{m^2} \)
\( \text{sphere radius} = 16/2 = 8\,\mathrm{cm} = 0.08\,\mathrm{m} \)
\( \displaystyle \frac{\varepsilon_0 \pi (0.02)^{2}}{d} = 4\pi\varepsilon_0 (0.08) \)
\( \displaystyle d = \frac{(0.02)^{2}}{4\times0.08} = \frac{4\times10^{-4}}{0.32} = 1.25\times10^{-3}\,\mathrm{m} = 1.25\,\mathrm{mm} \)