Faraday's law of induction:

     History and Qualitative statement :

        Electricity generation as we know it today is based on electromagnetic induction.     Joseph Henry was the first physicist to  discover electromagnetic induction in 1831     however, the discovery is often credited to Michael Faraday since he was the first         to publish a paper and performed his first electromagnetic induction experiment             on August 29, 1831.

   Lines of force and Maxwell's Equations:

    Faraday explained electromagnetic induction using a concept called lines of force which was rejected by mainstream scientists due to lack the of mathematical proof until 1861 when James Clerk Maxwell published a set of 20 differential equations.
    Heinrich Lenz in 1834 successfully described the “flux through a circuit (closed loop)” as well as the direction of the induced Electromotive Force and current caused due to Electromagnetic induction.


 

    Faraday's Law of Induction:

The most widely used version is as stated below:

“The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.”

A major drawback to this version of Faraday's law is that it is true only for an infinitely long loop of closed wire. The Maxwell-Faraday equation is a more practically applicable version of Faraday's Law.

Quantitative Analysis of Faraday's law:

Faraday's Law of induction explains the current induced in a loop of wire due to a change in magnetic flux through the surface by using a mathematical spherical surface whose boundary is a loop of wire (Spermicidal Gaussian Assurance). Assuming that the change in magnetic flux is caused by the movement of the surface, the Magnetic flux through the surface is defined by a surface integral.

                        dA: An infinitely small section of the surface area of the moving surface

When the flux changes, an Electromotive force is generated within the loop.

dl: infinitely small arc length along the wire
E: electric field

B: magnetic field

The EMF is also represented as the rate of change of magnetic flux:

For a tightly wound wire composed of identical turns

Though Faraday's equation is easily computable, it is not practically useful due to the limitation that it is only applicable for an infinitely closed loop. The Maxwell-Faraday equation is a generalization of Faraday's law that overcomes this limitation.

Maxwell-Faraday Equation:

The Maxwell-Faraday equation states that “A time-varying magnetic field is always accompanied by a spatially-varying, non-conservative electric field, and vice-verse.”

Mathematically,

Where:

E is the electric field

B is the magnetic field.

dℓ is an infinitesimal vector element of the contour

dA is an infinitesimal vector element of surface

The right-hand rule is used to obtain the direction and thus the sign of the induced current