Derivation of Maxwell's Third Equation

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Derivation of Maxwell's Third Equation

Faraday's Law

Faraday's law of electromagnetic induction is

E = -d_Bdt

The magnetic flux is defined as

_B = _S B dS

For a stationary closed circuit,

E = _C E dl

Therefore,

_C E dl = -ddt _S B dS

Application of Stokes' Theorem

According to Stokes' theorem,

_C E dl = _S (E) dS

Substituting into Faraday's law,

_S (E) dS = -ddt _S B dS

Since the surface is stationary,

_S (E) dS = -_S B t dS

Combining both sides,

_S ( E + B t ) dS =0

Because this holds for every arbitrary surface, the integrand must vanish.

Differential Form

Thus,

E + B t =0

Finally,

E = -B t