Displacement Current and Ampere’s law

Last updated on Saturday, September 30th, 2023

Displacement current you can understand by the time-varying electric field. Similar as in the Faraday’s law you have used the time-varying magnetic field to produce the EMF. Now the question is why do you need to define this displacement current and what is it? Is this similar to Alternating Current/Direct Current or different from these?

A need of Displacement Current

When Maxwell defined all the basic electrodynamics laws in differential from then it was observed that Ampere’s law is valid only for the direct currents not for the alternating current. As you know that we use Ampere’s law to study and find the magnetic field at a certain point due to the current in a conductor. Provided the current enclosed by the loop which is made by equidistant points from the conductor. At which you are studying the magnetic field.

For the direct current, it was studying by an old formula. But when a magnetic field was studied in between the two charge plates there was no method, like in between the plates of a capacitor.

Nature of displacement current

As I have said earlier that displacement current is the result of the time-varying electric field in between two electrodes. When alternating current passed through a circuit in which a capacitor is attached the ac changes the polarity of each electrode two times in a cycle. So in this way, the electric field in between two plates change the magnitude and direction accordingly. This time-varying electric field produces the magnetic field in between the two plates.

For a mathematical explanation, you can watch this video of displacement current and its form from the link in Hindi and English. The Displacement Current Density in Ampere’s Law you can watch to see its role in electrodynamics.

Displacement current nature is different from the AC/DC visualization. It is because of the varying electric field in between the two electrodes. So its nature is not like the alternating or direct current.

Inconsistency in the Ampere’s Law

Ampere’s law was valid for the direct current it was proved by using the continuity equation. Later it was modified for the alternating current in electrodynamics. The inconsistency in Ampere’s law was removed by introducing the displacement current.

For the derivation purpose and basic concept, you can watch this video Modification of Ampere’s Circuital Law and Displacement Current.

This video tutorial includes the formulation and basic concepts for the modified Ampere’s Circuital Law.

Part-1 Continuity equation

From the continuity equation, we have seen that if the current is DC, then divergence of the current density J will be zero. It means the magnitude of the current is constant at both the end (solenoidal condition).  The rate of the volume charge density is zero.

Part-2 Ampers’s law:

If we use ampers’ law in differential form and take the divergence we see the same situation as seen above in case of continuity equation, that divergence of J is zero, so obviously it means the present Ampere’s law was valid only for the DC current. so what about the Ampere’s law if the current is AC (Alternating Current). So this was the inconsistency in the Ampere’s law. How Maxwell remove it?
it removed by using the concept of displacement current.

Part-3

In this section, I have derived the Ampere’s law in differential form by using Stokes law. after that take the divergence which results discussed in the video. So the point was only to introduce the role of alternating current also in the Ampere’s circuital law.

Part-4

we use Ampere’s circuital law to calculate or determine the magnetic field which is produced by the associated current. The condition is only that current should be enclosed by the loop. (only the Ampere’s law will apply). If the current is flowing through the straight wire one can find the magnetic field at any point near or far from the conductor by considering the close loop…..but one more case is considered here i.e. the capacitor. Suppose one wanted to find the magnetic field at the center of capacitor plates, what will be the process for that? In this case, we see that current is not enclosed by taking any type of surface and as a result, the right-hand side of the Amper’s law will be zero and hence the magnetic field will be zero. It means there is no field in the case of direct current. But when you apply the alternating current here, capacitor works and as a result, the polarity on the plates change as with the cycle of the alternating current. Hence the electric field within the plates will change with time. (You can see here the time-varying electric field).

As you know in the case of Faraday’s law, the time-varying magnetic filed produce the electric field, similarly here in our case of the capacitor where an electric field is changing with time produce the magnetic field. It can be seen by the mathematical term of displacement current. There is the time-varying electric field which produces the magnetic field. So in the latest form with displacement current one can use either dc or ac to determine the magnetic field.