Faraday’s Law of Induction

Faraday’s law of induction is a fundamental law of electromagnetism that relates electromagnetic induction to changes in magnetic flux. In its familiar circuit form, it states that the induced electromotive force around a conducting loop equals the negative rate of change of the magnetic flux through the loop. In its field form, it describes the relationship between a time-varying magnetic field and the circulation of the electric field. These formulations underlie the operation of generators, transformers, and inductive circuit components. (openstax.org)

Magnetic flux and the circuit law

The magnetic flux through an oriented surface $S$ is

\Phi_B=\int_S\mathbf B\cdot d\mathbf A,

where $\mathbf B$ is the magnetic field and $d\mathbf A$ is an area element directed perpendicular to the surface. For a uniform field through a flat surface of area $A$ ,

\Phi_B=BA\cos\theta,

where $\theta$ is the angle between the field and the surface normal—not the plane of the surface. Flux can therefore change because the field changes, the enclosed area changes, or the loop changes orientation. (openstax.org)

For a single loop, the circuit law is

\boxed{\mathcal E=-\frac{d\Phi_B}{dt}}.

Here $\mathcal E$ is the induced emf. Despite its name, emf is not a mechanical force: it measures the integrated driving force per unit charge around a circuit and has units of volts. In the International System of Units, magnetic flux is measured in webers, with $1\,\mathrm{Wb/s}=1\,\mathrm V$ . (feynmanlectures.caltech.edu)

For a coil with a fixed number $N$ of series-connected turns, each linking the same flux,

\mathcal E=-N\frac{d\Phi_B}{dt}.

If the turns link different fluxes, the relevant quantity is their signed sum, rather than $N$ times an arbitrary single-turn flux. (openstax.org)

Direction: Lenz’s law

The negative sign expresses Lenz’s law: the induced emf has a direction such that the current it drives produces a magnetic field opposing the change in flux that caused the induction. The positive direction around the loop and the positive surface normal are related by the right-hand rule. (openstax.org)

For example, if an external magnetic field directed into a loop’s plane increases, the induced current produces a field directed out of the plane. If the external inward field decreases, the induced current instead produces an inward field, tending to sustain the diminishing flux. The induced field opposes the change, not necessarily the external field itself. (openstax.org)

This behavior is consistent with conservation of energy. When a conducting rod is pulled through a magnetic field as part of a closed resistive circuit, the induced current produces a magnetic force opposing the motion. The external work required to maintain the motion supplies the electrical energy dissipated by the circuit. (openstax.org)

Integral and differential field forms

For a closed curve $C$ and a spanning surface $S$ , both fixed in space, the Maxwell–Faraday equation is

\boxed{ \oint_C\mathbf E\cdot d\boldsymbol\ell = -\frac{d}{dt}\int_S\mathbf B\cdot d\mathbf A }.

The left side is the line integral of the electric field around the curve. The curve need not be a physical wire: the equation also applies to an imaginary closed path in empty space. (feynmanlectures.caltech.edu)

Its local differential form is

\boxed{\nabla\times\mathbf E =-\frac{\partial\mathbf B}{\partial t}}.

The curl measures the local circulation of a vector field. Stokes’ theorem connects this differential equation to the integral form. The Maxwell–Faraday equation is one of Maxwell’s equations, which describe classical electric and magnetic fields. (feynmanlectures.caltech.edu)

Unlike an electrostatic field, an induced electric field can have a nonzero integral around a closed path. Thus it cannot, in general, be represented solely as the negative gradient of a single-valued scalar electric potential. (feynmanlectures.caltech.edu)

Stationary and moving circuits

Induction involves two distinguishable contributions:

Transformer emf: in a stationary circuit, changing magnetic flux is associated with a circulating electric field that drives charges.
Motional emf: in a moving conductor, charges experience the magnetic part of the Lorentz force. (feynmanlectures.caltech.edu)

For a moving or deforming material loop, these contributions combine as

\mathcal E = \oint_{C(t)} \left(\mathbf E+\mathbf u\times\mathbf B\right) \cdot d\boldsymbol\ell = -\frac{d}{dt}\int_{S(t)}\mathbf B\cdot d\mathbf A,

where $\mathbf u$ is the local velocity of the circuit material. The derivative on the right includes changes in both the field and the moving surface. (feynmanlectures.caltech.edu)

A standard example is a conducting rod of length $\ell$ sliding along rails at speed $v$ . If the rod, its velocity, and a uniform magnetic field are mutually perpendicular, the magnitude of the motional emf is

|\mathcal E|=B\ell v.

For a closed circuit with resistance $R$ , neglecting inductive transients, Ohm’s law gives $I=B\ell v/R$ . A moving isolated rod can also develop a potential difference between its ends without carrying a sustained current. (openstax.org)

Historical development

Michael Faraday demonstrated electromagnetic induction at the Royal Institution on August 29, 1831, using two coils wound on an iron ring. Establishing current in one coil produced a brief current in the other. The apparatus demonstrated induction between electrically separate circuits sharing a magnetic structure. (rigb.org)

James Clerk Maxwell subsequently incorporated Faraday’s experimental discovery into a mathematical field theory. The modern differential equation expresses induction as a local relationship between electric and magnetic fields, rather than only as a rule for wire circuits. (feynmanlectures.caltech.edu)

Applications

Generators and motor back emf

An electric generator converts mechanical input into electrical output through induction. For an ideal coil of $N$ turns and area $A$ , rotating with angular speed $\omega$ in a uniform field,

\Phi_B=BA\cos(\omega t), \qquad \mathcal E=NBA\omega\sin(\omega t).

The changing orientation produces an alternating emf, providing a basic model of alternating-current generation. Rotation in an electric motor likewise produces a back emf opposing the applied electrical drive. (openstax.org)

Transformers

An electrical transformer uses changing magnetic flux to couple primary and secondary windings. When both windings link the same flux and losses are neglected, their voltage magnitudes satisfy

\frac{V_s}{V_p}=\frac{N_s}{N_p}.

The turns ratio determines whether voltage is increased or decreased. A constant direct current does not provide continuing transformer action after the switching transient: the linked flux must vary with time. (openstax.org)

Self-inductance

A changing current also changes the magnetic flux linked by its own circuit. For a fixed, linear circuit with self-inductance $L$ ,

N\Phi_B=LI, \qquad \mathcal E_{\mathrm{self}}=-L\frac{dI}{dt}.

The induced emf opposes changes in current, not current itself. If $L$ varies, the more general expression is $-d(LI)/dt$ . This behavior is deliberately used in inductors and is also present in ordinary wiring. (openstax.org)

Eddy currents

Induced currents can circulate within bulk conductors, rather than only in discrete wire loops. These eddy currents enable magnetic damping, electromagnetic braking, and metal detection. They also cause unwanted losses in transformer cores; insulated laminations restrict the available current paths and reduce those losses. (openstax.org)

Scope and qualifications

Faraday’s law determines induced emf, not the complete current response. The resulting current also depends on resistance, inductance, and the rest of the circuit. In particular, $\mathcal E/R$ is not a general substitute for solving a circuit with significant inductive dynamics. (openstax.org)

The moving-loop flux rule requires careful identification of the material circuit. In devices with sliding contacts or extended conducting bodies, such as a rotating conducting disk, a geometrically chosen “circuit” may not move with the conductor. An apparent failure of the simple flux calculation is not a failure of the Maxwell–Faraday equation: the electric-field circulation and magnetic Lorentz-force contribution must be evaluated for the actual charge path. (feynmanlectures.caltech.edu)

References

1 Faraday’s Law - University Physics Volume 2openstax.org
2 Faraday’s Law of Induction: Lenz’s Law - College Physics 2eopenstax.org
3 Motional Emf - University Physics Volume 2openstax.org
Michael Faraday’s ring-coil apparatusrigb.org
6 Electric Generators and Back Emf - University Physics Volume 2openstax.org
7 Transformers - College Physics 2eopenstax.org
2 Self-Inductance and Inductors - University Physics Volume 2openstax.org
5 Eddy Currents - University Physics Volume 2openstax.org