Faraday's law of induction

Magnetic flux

Electromagnetic induction begins with magnetic flux. For a uniform magnetic field through a flat area,

Phi_B=ec{B}cdotec{A}=BAcos heta

where $heta$ is the angle between the magnetic field and the area vector. For a nonuniform field or curved surface,

Phi_B=int ec{B}cdot dec{A}

Flux measures how much magnetic field passes through a surface. It can change if the field changes, the area changes, or the angle changes.

Faraday's law

Faraday's law states that changing magnetic flux through a circuit induces an emf:

mathcal{E}=-rac{dPhi_B}{dt}

For a coil with $N$ turns,

mathcal{E}=-Nrac{dPhi_B}{dt}

The negative sign represents Lenz's law, which gives the direction of the induced emf.

An emf is energy per unit charge supplied around a circuit. It can drive current if a conducting path is present.

Induced electric fields

Faraday's law is deeper than battery-like voltage. A changing magnetic field creates a circulating electric field. The general integral form is

oint ec{E}cdot dec{ell}=-rac{dPhi_B}{dt}

This electric field is nonconservative. Unlike electrostatic fields, its line integral around a closed loop is not zero when magnetic flux changes.

Motional emf

Induction can also occur when a conductor moves through a magnetic field. Charges in the conductor experience magnetic force

ec{F}=qec{v} imesec{B}

For a rod of length $L$ moving at speed $v$ perpendicular to a magnetic field, the motional emf is

$mathcal{E}=BLv$

This happens because magnetic forces separate charges along the rod.

Generator principle

An electric generator uses changing magnetic flux to produce emf. A loop rotating in a magnetic field has flux

$Phi_B=BAcos(omega t)$

Faraday's law gives

$mathcal{E}=BAomegasin(omega t)$

For $N$ turns,

$mathcal{E}=NBAomegasin(omega t)$

This is the basis of alternating current generation in power plants.

Transformers

Transformers use changing magnetic flux to transfer energy between coils. An alternating current in one coil creates changing magnetic flux in a core, inducing emf in another coil. The ideal transformer voltage ratio is

rac{V_s}{V_p}=rac{N_s}{N_p}

where subscripts refer to secondary and primary coils.

Transformers work only with changing currents, not steady DC in ideal operation.

Faraday's law and circuits

If a loop has resistance $R$, an induced emf drives current

I=rac{mathcal{E}}{R}

The direction of current follows Lenz's law. The induced current produces its own magnetic field that opposes the flux change.

The big idea

Faraday's law says changing magnetic flux produces emf and circulating electric fields. Flux can change through changing field, area, angle, or motion. Induction underlies generators, transformers, inductors, wireless charging, electric guitars, and much of modern electrical technology.