The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit."Faraday's Law, which states that the electromotive force around a closed path is equal to the negative of the time rate of change of magnetic flux enclosed by the path" . Mathematically,

${\displaystyle \epsilon =-N{\frac {dB}{dt}}=-NL{\frac {dI}{dt}}}$

This version of Faraday's law strictly holds only when the closed circuit is a loop of infinitely thin wire

Electromagnetic induction

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


Faraday's law of induction makes use of the magnetic flux ΦB through a hypothetical surface Σ whose boundary is a wire loop. Since the wire loop may be moving, we write Σ(t) for the surface. The magnetic flux is defined by a surface integral:

${\displaystyle \Phi _{B}=\iint \limits _{\Sigma (t)}\mathbf {B} (\mathbf {r} ,t)\cdot d\mathbf {A} \ ,}$

where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field, and B·dA is a vector dot product (the infinitesimal amount of magnetic flux). In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.

When the flux changes—because B changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an EMF, ${\displaystyle {\mathcal {E}}}$, defined as the energy available from a unit charge that has travelled once around the wire loop. Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads. According to the Lorentz force law (in SI units),

${\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}$

the EMF on a wire loop is:

${\displaystyle {\mathcal {E}}={\frac {1}{q}}\oint _{\mathrm {wire} }\mathbf {F} \cdot d{\boldsymbol {\ell }}=\oint _{\mathrm {wire} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\cdot d{\boldsymbol {\ell }}}$

where E is the electric field, B is the magnetic field (aka magnetic flux density, magnetic induction), d is an infinitesimal arc length along the wire, and the line integral is evaluated along the wire (along the curve the conincident with the shape of the wire).

The EMF is also given by the rate of change of the magnetic flux:

${\displaystyle {\mathcal {E}}=-{{d\Phi _{B}} \over dt}\ }$,

where ${\displaystyle {\mathcal {E}}}$ is the electromotive force (EMF) in volts and ΦB is the magnetic flux in webers. The direction of the electromotive force is given by Lenz's law.

For a tightly wound coil of wire, composed of N identical turns, each with the same ΦB, Faraday's law of induction states that[1][2]

${\displaystyle {\mathcal {E}}=-N{{d\Phi _{B}} \over dt}}$

where N is the number of turns of wire and ΦB is the magnetic flux in webers through a single loop.

Application

1. Ampere's law sates that magnetic field strength is proportional to its inductance and current flow through it

${\displaystyle +B=+LI}$ . Ampere's law


2. Lorentz's law states that change in current with changing in time produces potential difference

${\displaystyle +V=+{\frac {dB}{dt}}=+L{\frac {dI}{dt}}}$ . Lorentz's law


3. Lenz's law states that induced current is generated from current but opposite in direction

${\displaystyle -\phi =-NB=-NLI}$ . Lenz's law


4. Faraday law states that induced voltage is generated from changing induced current with change in time

${\displaystyle -\epsilon =-{\frac {d\phi }{dt}}=-NL{\frac {dI}{dt}}}$ . Faraday's law


Reference

1. Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1
2. Nave, Carl R. "Faraday's Law". HyperPhysics. Georgia State University. Retrieved 29 August 2011.