Question 12: Describe the statement of four Maxwell’s equation. Use Maxwell’s theory to show that
Maxwell in 1884 formulated the basic principles of electromagnetism in terms of four equations called Maxwell’s equations. These equations give us information about the existence and production of electromagnetic waves.
For a changing magnetic flux density B through a certain region of space produces an induced emf in the region. This induced emf ϵ induces a current flowing through a closed loop of wire in the region. Applying Faraday’s law
The electric field E will be circular in direction and its magnitude is constant for the loop. If a unit positive charge is circulated around the loop then in one cycle the work done is
Here r is the radius and 2πr is the circumference (total distance) of the loop. Since the electric force for the electric field intensity E is Fe=qE, hence
Now the electromotive force is the work done on a positive unit charge, therefore
Compare equations (1) and (2)
This is the first result for the induced electric field.
(2) In the same way, an electric flux gives rise to a magnetic field. Take the example of a capacitor connected to a battery. Initially the current is rising in the circuit and then decreases to zero when the capacitor is fully charged. Then there is an electric field between the plates of the capacitor. If Q is the charge on the capacitor, C is its capacitance, A is the area of the plate and d their mutual separation then
However, when the capacitor is connected to an alternating source, the current is continuously flowing in the circuit. We observe that when the current from the alternating source changes, the electric field between the plates also changes. The flow of current between the plates is due to the changing electric field between them. In the outer circuit, the current flow is due to the conduction of electrons in the wire. This was first observed by Maxwell that a change in electric field also causes a current flow. Therefore, using the definition of current and the above equation,
Φ is the electric flux through area A.
This equation shows that a changing electric flux produces the same effect as the current. Therefore, a changing electric flux is equivalent to the current. This is called displacement current as against the conduction current which is due to the flow of electrons.
This is the second result required in the question.