Question

A parallel - plat ecapacitor having plate area A and plate separation d is joined to a battery of emf $ϵ$ and internal resistance R at t = 0. Consider a plane surface fo area $\frac{A}{2}$ parallel to the plates and situated symmetrically between them. Find the displacement. current through this surface as a function of time.

Answer 1

We know $E=\frac{Q}{{ϵ}_{0}A}$ (Electric field)
$\varphi =E.A=\frac{Q}{{ϵ}_{0}A}.\frac{A}{2}=\frac{Q}{{ϵ}_{0}2}{i}_0={ϵ}_0\frac{{\varphi }_E}{dt}={ϵ}_0\frac{d}{dt}\left(\frac{Q}{{ϵ}_{0}2}\right)=\frac{1}{2}\left(\frac{dQ}{dt}\right)=\frac{1}{2}\frac{d}{dt}{(-\frac{1}{RC})}^{\frac{e^{-t}}{RC}}=-\frac{E}{2R}{e}^{\frac{-td}{ϵAR}}$