30.9:
Displacement Current
Ampere's law states that the line integral of a magnetic field along a closed curve equals the permeability of free space times the net current passing through the loop.
Consider a parallel plate capacitor connected with a battery in a charging condition.
Applying Ampere's law to the shown Amperian loop with two surfaces gives two different magnetic field values, which is impossible.
As the capacitor is charging, the electric field, and hence the electric flux, increases through the bulging surface. The value of electric flux can be obtained in terms of the charge.
In 1865, James Clerk Maxwell, predicted that due to this time-varying electric field, a non-zero magnetic field is produced between the plates of the capacitor through a fictitious current called displacement current.
The expression for displacement current is given in terms of electric flux.
Thus, Ampere's law is modified with the inclusion of an additional term for displacement current. It is known as the Ampere-Maxwell law or generalized Ampere's law.
30.9:
Displacement Current
Ampère's law, in its usual form, does not work in places where the current changes with time and is not steady. Thus, Maxwell suggested including an additional contribution, called the displacement current, Id, to the real conduction current I.
where the displacement current is defined to be
Here, ε0 is the permittivity of free space, and ΦE is the electric flux.
The displacement current is an extra term in Maxwell's equations that is analogous to a real current in Ampère's law. However, it is produced by a changing electric field. It accounts for a changing electric field producing a magnetic field, just as a real current does, but the displacement current can produce a magnetic field even when no real current is present. When this extra term is included, the modified Ampère's law equation becomes
In this way, Ampère's law can be modified so that it works in all situations, and it is independent of the surface through which the current I is measured.
Suggested Reading
OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves