Physics with Calculus/Electromagnetism/Electromagnetic Induction

Electromagnetic induction is the production of voltage across a conductor moving through a magnetic field.

Michael Faraday is generally credited with the discovery of the induction phenomenon in 1831. Around 1830 to 1832 Joseph Henry made a similar discovery, but did not publish his findings until later.

Details

Faraday found that the electromotive force (EMF) produced around a closed path is proportional to the rate of change of the magnetic flux through any surface bounded by that path.

In practice, this means that an electric current will be induced in any closed circuit when the magnetic flux through a surface bounded by the conductor changes. This applies whether the field itself changes in strength or the conductor is moved through it.

Electromagnetic induction underlies the operation of electrical generator, all electric motors, transformers, induction motors, synchronous motors, solenoids, and most other electrical machines.

Faraday's law of induction|Faraday's law of electromagnetic induction states that:

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

Thus:

${\mathcal {E}}$ is the electromotive force (emf) in volts

Φ_B is the magnetic flux in Weber (Wb)|webers

For the common but special case of a coil of wire, composed of N loops with the same area, Faraday's law of electromagnetic induction states that

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

where

${\mathcal {E}}$ is the electromotive force (emf) in volts

N is the number of turns of wire

Φ_B is the magnetic flux in Weber (Wb)|webers through a single loop.

A corollary of Faraday's Law, together with Ampere's and Ohm's laws is Lenz's law:

The emf induced in an electric circuit always acts in such a direction that the current it drives around the circuit opposes the change in magnetic flux which produces the emf.

The direction mentioned in Lenz's law can be thought of as the result of the minus sign in the above equation.

Example of conducting loop spinning in magnetic field

Faraday's law of induction|Faraday's law of electromagnetic induction states that the induced electromotive force is the negative time rate of change of magnetic flux through a conducting loop.

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

where ${\mathcal {E}}$ is the electromotive force (emf) in volts and Φ_B is the magnetic flux in Weber (Wb)|webers. For a loop of constant area, A, spinning at an angular velocity of $\omega$ in a uniform magnetic field, B, the magnetic flux is given by

$\Phi _{B}=B\cdot A\cdot \cos(\theta ),$

where θ is the angle between the normal to the current loop and the magnetic field direction. Since the loop is spinning at a constant rate, ω, the angle is increasing linearly in time, θ=ωt, and the magnetic flux can be written as

$\Phi _{B}=B\cdot A\cdot \cos(\omega t).$

Taking the negative derivative of the flux with respect to time yields the electromotive force.

${\mathcal {E}}=-{\frac {d}{dt}}\left[B\cdot A\cdot \cos(\omega t)\right]$	Electromotive force in terms of derivative
$=-B\cdot A{\frac {d}{dt}}\cos(\omega t)$	Bring constants (A and B) outside of derivative
$=-B\cdot A\cdot (-\sin(\omega t)){\frac {d}{dt}}(\omega t)$	Apply chain rule and differentiate outside function (cosine)
$=B\cdot A\cdot \sin(\omega t){\frac {d}{dt}}(\omega t)$	Cancel out two negative signs
$=B\cdot A\cdot \sin(\omega t)\omega$	Evaluate remaining derivative
$=\omega \cdot B\cdot A\sin(\omega t).$	Simplify.