
According to Ampere’s circuital law, the line integral of the magnetic field around any closed path or circuit is equal to μ_{0 }times the total current

In the calculation of magnetic field, Ampere’s circuital law brought in several contradictions. When a different surface was used to find the magnetic field, the result was different.

It was concluded that a term was missing from the Ampere’s circuital law—the electric field. This electric field passes the surfaces between the plates of the capacitor used.

Each plate of a capacitor has an area A, and a total charge Q. Then, the magnitude of the electric field E is,

Using Gauss’ law, the electric flux Φ_{E} through the surface is calculated as

As the charge Q on the capacitor plates changes with time, there is a current,
i = dQ/dt
This is the missing term in Ampere’s circuital law.

The current carried by conductors due to the flow of charges is called conduction current, and the current (new term) due to the changing electric field is called displacement current or Maxwell’s displacement current.

Total current, i = Conduction current (i_{c}) + Displacement current (i_{d})

Outside the capacitor plate, i = i_{c}, and inside the capacitor plate, i = i_{d}

Ampere−Maxwell law is given as
The total current passing through any surface, of which the closed loop is the perimeter, is the sum of the conduction and displacement current.
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