Question

Define electric capacity. Derive an expression for the capacitance of a parallel plate capacitor in which a dielectric medium of dielectric constant $K$ fills the space between the plates. Draw the necessary diagram.

Answer 1

Capacitance or electric capacity indicates the ability of a system to store charge. It is defined as the ratio of charge stored in the conductors to the potential difference across the conductors. Its SI unit is Farad ($F$). Hence:

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

To derive the expression of capacitance of parallel plate capacitor of conductor area $A$ and distance between the conductors as $d$ with a dielectric of dielectric constant $K$ carrying a charge $Q$ and potential difference $V$, refer to the attached figure.

Electric field between plates of the capacitor can be found using Gauss Law.

$E=\frac{\sigma }{K{\epsilon }_o}$

$E=\frac{Q}{K{\epsilon }_{o}A}$ as $\sigma =\frac{Q}{A}$

Electric field is defined as the gradient of potential.

$E=\frac{V}{d}$

$⟹V=\frac{Qd}{K{\epsilon }_{o}A}$

By definition of capacitance,

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

$=\frac{K{\epsilon }_{o}A}{d}$