Question

A circular parallel plate capacitor has radius $r$ . Air is filled between the plates if the capacitance of capacitor is equal to that of the sphere of radius $R$ , then what is the distance between the plates ?

Answer 1

Capacitance of parallel plate capacitor :
$C=\frac{{\epsilon }_{0}A}{d}=\frac{{\epsilon }_0\pi r^2}{d}$
Capacitance of spherical capacitor :
$C=4\pi {\epsilon }_{0}R$
According to question
$4\pi {\epsilon }_{0}R=\frac{{\epsilon }_0\pi r^2}{d}$
$d=\frac{r^2}{4R}$