Q. 1

# Show that the dimensions of the displacement current are that of an electric current.

Answer :

The displacement current I_{d} is produced by a varying electric

field. It is given by the relation

where ϵ_{0} is the electric permittivity of free space(vacuum) and is

equal to 8.85 × 10^{-12} C^{2} N^{-1} m^{-2} and *ϕ*_{E} is the electric flux produced

by the time varying electric field.

To find the dimension of I_{d} and check whether it is the same as that

of electric current, we need to find the dimension of _{.}

We can simplify using Gauss’s Law. According to Gauss’s law

the electric flux *ϕ*_{E} through a surface is given as

Using the above relation, the displacement current becomes

The dimension of displacement current is the dimension of the

quantity in the R.H.S of the above equation

As [A] is the dimension of electric current so displacement current

has same dimension as that of electric current.

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Displacement current goes through the gap between the plates of a capacitor when the charge of the capacitor

HC Verma - Concepts of Physics Part 2Consider 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)

HC Verma - Concepts of Physics Part 2

You are given a 2μF parallel plate capacitor. How would you establish an instantaneous displacement current of 1mA in the space between its plates?

Physics - ExemplarUsing B = μ_{0} H find the ratio E_{0}/H_{0} for a plane electromagnetic wave propagating through vacuum. Show that is has the dimensions of electric resistance. This ratio is a universal constant called the impedance of free space.

Show that the dimensions of the displacement current are that of an electric current.

HC Verma - Concepts of Physics Part 2