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Maxwell third equation

Also called the Faraday law of "electromagnetic induction". The Maxwell third equation has two statements.

Statement I: 
It states that whenever a magnetic flux link with a circuit changes, then induced electromotive force (emf)  is set up in the circuit.
Statement II:
The magnitude of induced emf is equal to the rate of magnetic flux linked with the circuit.

Integral form:
Therefore;
$$Induced-emf = - \frac {d\psi_m}{dt}$$
where,
$$\psi_m = \int B.ds$$   ......(5)
The negative sign is because of Lentz law,  which states that the induced emf set up a current in such a direction that the magnetic effect produced by it opposes the cause producing it.
Also, the definition of emf states that the emf is the closed line integral of the non conservative electric field generated by the battery.
That is:
$$emf = \int E.dL$$ ........(6)
Comparing equ(5) and equ(6) we have:
$$\int E.dL = -\int_{s} \frac{dB.ds}{dt}$$ ...(7)

Differential form
Applying Stokes theorem to the LHS of equ(7) to change line integral to surface integral.
That is,
$$\int E.dL = \int (\nabla X E).ds$$ ........(*)
Substituting equ(*) into equ(7):
$$\int_{s} (\nabla X E) ds = -\int_{s} \frac {d}{dt} (B.ds)$$
Since, two surface integrals are equal only when their integrands are equal.
Thus,
$$\nabla X E = - \frac{dB}{dt}$$ ..........(8)
Equ(8) is the differential form of the Maxwell equation.

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