Consider the situation of the previous problem. Define displacement resistance R_d = V/i_d of the space between the plates where V is the potential difference between the plates and i_d is the displacement current. Show that R_d varies with time as

R_d = R (e^t/τ – 1)

Given: The displacement resistance

We will first calculate the displacement current.

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.

We need to calculate the electric flux through the capacitor plate.

As the charge on the capacitor plate at time t can be taken as Q so

by using Gauss’s law, we will calculate the electric flux.

According to Gauss’s law

so the electric flux would be

.

As the capacitor is charging, the charge will be a function of time

given as

where C is the capacitance of the capacitor, V is the potential drop

at time t, R is the series resistance and V₀ is the potential at time

t=0. Now the flux is .

Now by our definition, the displacement current is given by

which is

The displacement current is

We know that

therefore , where τ is the time constant.