A capacitor, made of two parallel plates each of plate are A and separation d, is being charged by an external a.c. source. Show that the displacement current inside the capacitor is the same as the current charging the capacitor.

Explanation

Let a parallel plate capacitor of capacitance `C=(in_(0)A)/(d)` is being charged by an external source whose voltage changes sinusoidally as `V=V_(m)sin omegat`.
Then instantaneous charge on capacitor plate `q=CV=CV_(m)sin omegat`
`therefore" Conduction current"`
`I_(C )=(dq)/(dt)=(d)/(dt)(CV_(m)sin omegat)=CB_(m)omega cos omega t" ...(i)"`
In the free space between the plates of capacitor a displacement current `I_(D)` is set up, whose magnitude is given as :
`I_(D)=in_(0)(dphi_(E))/(dt)=in_(0)(d)/(dt)(EA)=in_(0)(d)/(dt)((V)/(d)A)=(in_(0)A)/(d)(dV)/(dt)=C(dV)/(dt)=C(dV)/(dt)=C(d)/(dt)=C(d)/(dt)(V_(m)sin omegat)`
`=CV_(m)cos omega t" ...(ii)"`
Comparing (i) and (ii), we find that `I_(C)=I_(D)` i.e., the displacement current inside the capacitor is the Comparing (i) and (ii), we find that `I_(C )=I_(D)` i.e., the displacement current inside the capacitor is the same as the current charging the capacitor.