Displacement current is a concept in electromagnetism that was introduced by James Clerk Maxwell in 1865. It is a quantity that appears in Maxwell's equations, which describe the behavior of electric and magnetic fields. Displacement current plays a crucial role in understanding electromagnetic induction and the generation of electromagnetic waves.



Displacement current is the flow of electric charge that occurs in a region where there is a changing electric field. It is not an actual current of moving charges like conduction current, but rather a mathematical construct that helps explain the behavior of electric and magnetic fields.



To derive the concept of displacement current, let's consider a situation where a time-varying electric field exists between the plates of a capacitor. The electric field between the plates changes with time, inducing a changing magnetic field according to Faraday's law of electromagnetic induction.

1. Faraday's Law: According to Faraday's law, a changing magnetic field induces an electric field. This means that a time-varying magnetic field can generate an electric field.

2. Amperian Loop: Consider an Amperian loop that encloses the plates of the capacitor. If there were only conduction current, the Ampere's law would state that the magnetic field integrated along the loop is equal to the conduction current enclosed by the loop.

3. Maxwell's Addition: Maxwell added an additional term to Ampere's law to account for the changing electric field. He introduced the concept of displacement current, denoted by the symbol Id, which is proportional to the rate of change of the electric field:

∮B·dl = μ₀(Ic + Id)

Here, B is the magnetic field, dl is an infinitesimal length element along the Amperian loop, μ₀ is the permeability of free space, Ic is the conduction current, and Id is the displacement current.

4. Consistency with Conservation of Charge: The concept of displacement current was introduced to ensure the conservation of charge. Without displacement current, the equation would not be consistent with the principle of charge conservation.

5. Electromagnetic Waves: The concept of displacement current is essential for understanding the generation of electromagnetic waves. When a time-varying electric field exists, it gives rise to a changing magnetic field, which in turn generates a changing electric field. This self-sustaining process leads to the propagation of electromagnetic waves.



The derivation of displacement current helps us understand the connection between changing electric and magnetic fields. It provides a complete description of electromagnetic phenomena and is a fundamental concept in electromagnetism. The concept of displacement current is crucial for explaining electromagnetic induction, the behavior of capacitors, and the generation of electromagnetic waves.