Ampere's Circuital Law Modification | apni Physics
Channel: V H Apni Physics
Date: 2015-09-21

http://www.apniphysics.com/displacement-current-ampere-law/
This lecture includes the formulation for the modified Ampere's Circuital Law. 
1. Part
From the continuity equation we have seen that if the nature of current is DC, then divergence of the current density j is zero. It means the magnitude of current is constant at both the end (sloenoidal condition). 
2. Ampers's law:
If we use ampers's 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 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.

3. Part
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. 

4.Part
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 want 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 capacitor where electric field is changing with time produce the magnetic field. It can be seen by the mathematical term of displacement current. There is 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.