Maxwell third equation and its derivation

Statement.(a)  It states that,whenever magnetic flux linked with a circuit changes then induced electromotive force (emf) is set up in the circuit. This induced emf lasts so long as the change in magnetic flux continues.

(b) The magnitude of induced emf is equal to the rate of change of magnetic flux linked with the circuit.

Therefore                               induced emf= – dφm/dt

Where                                    φm=∫B.dS (5)

Here negative sign is because of Lenz’s 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 definition of emf states that emf is the closed line integral of the non-conservative electric field generated by the battery.

That is                   emf=∫E.dL                                                       (6)

Comparing equations (5) and (6), we get

∫E.dL= – ∫s d/d t B.dS                                                                    (7)

Equation (7) is the integral form of Maxwell’s third Equation or Faraday’s law of electromagnetic induction.

Note: You can also read the discussion and derivation of Maxwell first and second equation.

Differential form :

Apply Stoke’s theorem to L.H.S. of equations (7) to change line integral to surface integral.

That is                                   ∫ E.dL=∫(  ∇ x E).dS

By substituting above equation in equation(7), we get

∫­s (   ∇ x E).dS  = -∫ d/d t(B.dS)

As two surface integral are equal only when their integrands are equal.

Thus                         ∇ x E= – dB/ dt                                   (8)

Equation (8) is the Differential form of Maxwell’s third equation.

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