Displacement Current

We know that an electric current produces a magnetic field around it. J.C. Maxwell showed that for logical consistency, a changing electric field must also produce a magnetic field. Further, since magnetic fields have always been associated with currents, Maxwell postulated that this current was proportional to the rate of change of the electric field and called it displacement current. In this article, we will look at displacement current in detail.

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How a changing electric field produces a magnetic field?

To determine this, let’s look at the process of charging a capacitor. Further, we will apply Ampere’s circuital law to find a magnetic point outside the capacitor.

The figure above shows a parallel plate capacitor connected in a circuit through which a time-dependent current i_(t) flows. We will try to find the magnetic field at a point P, in the region outside the capacitor.

Consider a plane circular loop of radius r centred symmetrically with the wire. Also, the plane of the loop is perpendicular to the direction of the current carrying wire. Due to the symmetry, the magnetic field is directed along the circumference of the loop and has similar magnitude at all points on the loop.

However, as shown in the Figure(2) above, when the surface is replaced by a pot-like surface where it doesn’t touch the current but has its bottom between the capacitor plates or a tiffin-shaped surface (without the lid) and Ampere’s circuital law is applied, certain contradictions arise.

These contradictions arise since no current passes through the surface and Ampere’s law does not take that scenario into consideration. This leads us to understand that there is something missing in the Ampere’s circuital law. Also, the missing term is such which enables us to get the same magnetic field at point P regardless of the surface used.

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Maxwell’s Displacement Current

If we look at the last figure again, we can observe that the common thing that passes through the surface and between the capacitor plates is an electric field. This field is perpendicular to the surface, has the same magnitude over the area of the capacitor plats and vanishes outside it.

Hence, the electric flux through the surface is Q/ε₀ (using Gauss’s law). Further, since the charge on the capacitor plates changes with time, for consistency we can calculate the current as follows:

i = ε₀ (dQ/dt)

This is the missing term in Ampere’s circuital law. In simple words, when we add a term which is ε₀ times the rate of change of electric flux to the total current carried by the conductors, through the same surface, then the total has the same value of current ‘i’ for all surfaces. Therefore, no contradiction is observed if we use the Generalized Ampere’s Law.

Hence, the magnitude of B at a point P outside the plates is the same at a point just inside. Now, the current carried by conductors due to the flow of charge is called ‘Conduction current’. The new term added is the current that flows due to the changing electric field and is called ‘Displacement current’ or Maxwell’s Displacement current’.

Displacement Current Explained

By now we understand that there are two sources of a magnetic field:

1. Conduction electric current due to the flow of charges
2. Displacement current due to the rate of change of the electric field

Hence, the total current (i) is calculated as follows: (where i_c – conduction current and i_d – displacement current)

i = i_c + i_d

= i_c + ε₀(dQ/dt)

This means that –

• Outside the capacitor plates: i_c=i and i_d=0
• Inside the capacitor plates: i_c=0 and i_d=i

So, the generalized Ampere’s law states:

The total current passing through any surface of which the closed loop is the perimeter is the sum of the conduction current and the displacement current.

This is also known as – Ampere-Maxwell Law. It is important to remember that the displacement and conduction currents have the same physical effects. Here are some points to remember:

• In cases where the electric field does not change with time, like steady electric fields in a conducting wire, the displacement current may be zero.
• In cases like the one explained above, both currents are present in different regions of the space.
• Since a perfectly conducting or insulating medium does not exist, in most cases both the currents can be present in the same region.
• In cases where there is no conduction current but a time-varying electric field, only displacement current is present. In such a scenario we have a magnetic field even when there is no conduction current source nearby.

Faraday’s Law of Induction and Ampere-Maxwell Law

According to Faraday’s law of induction, there is an induced emf which is equal to the rate of change of magnetic flux. Since emf between two points is the work done per unit charge to take it from one point to the other, its existence simply implies the existence of an electric field. Rephrasing Faraday’s law:

A magnetic field that changes with time gives rise to an electric field.

Hence, an electric field changing with time gives rise to a magnetic field. This is a consequence of the displacement current being the source of the magnetic field. Hence, it is fair to say that time-dependent magnetic and electric fields give rise to each other.

Solved Examples for You

Question: Explain the missing term in Ampere’s circuital law

Solution: According to Ampere’s circuital law, the integral of magnetic field density (B) along an imaginary closed path is equal to the product of current enclosed by the path and permeability of the medium. However, upon passing a time-dependent current and changing the surface which does not touch the current, certain contradictions arise since Ampere’s law does not take these parameters into consideration. The inability of the law to determine the magnetic field at a point outside the region of the capacitor explains the missing term in the equation.