A parallel-plate capacitor 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 of area A/2, parallel to the plates and

situated symmetrically between them. Find the displacement current

through this surface as a function of time.

Given: Area of capacitor plates=A

Separation between the plates=d

Emf of the battery = ϵ

Internal resistance of the battery = R

Area of plane surface= A/2

Displacement current is the current which is generated by a time

varying electric field, not by the flow of charge carriers.

This current is also responsible for the generation of a time varying

magnetic field. The displacement current I_d is generated due to the

fact that the charge on capacitor plates is changing with time.

The displacement current is given by

where ϕ_E is the time varying electric flux through the plane surface

and ϵ₀ is the electric permittivity of free space(vacuum) and is equal

to 8.85 × 10^-12 C² N^-1 m^-2.

The electric field in the space between the plates can be given by

Guass’s Law. If the charge on the capacitor plate is Q and the area

of the plate is A(given), then by Guass’s law,

where E is the electric field and ϵ₀ is the electric permittivity of free

space and dS is a small area element on the plate.

Further (because the area vector

and electric field lines are both normal to the surface and in

same direction i.e. θ=0^° so cos θ=1)

So → , the electric field between the plates is .

This electric field produces and electric flux through the plane

surface given by

(because the area vector and electric field lines are both normal to

the surface and in same direction i.e. θ=0^° so cos θ=1)

Now the charge on the capacitor is changing with time as it is

charging. If the capacitance of the capacitor is C, then the charge Q

at time t will be

where ϵ is the potential between plates which is equal to the emf of battery and R is the resistance attached in series.

The displacement current I_d is given as

Thus the displacement current as a function of time is .