Faraday's law of induction/ Maxwell–Faraday equation is a generalization of Faraday's law, and forms one of Maxwell's equations..

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Faraday's law of induction

is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.^[1]^[2]

The Maxwell–Faraday equation is a generalization of Faraday's law, and forms one of Maxwell's equations.

1 History
2 Faraday's law
3 Proof of Faraday's law
4 "Counterexamples" to Faraday's law
5 Faraday's law and relativity
- 5.1 Two phenomena
- 5.2 Einstein's view
6 See also
7 References
8 Further reading
9 External links

Faraday's law

Qualitative statement

The most widespread version of Faraday's law states:

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.^[14]^[15]

This version of Faraday's law strictly holds only when the closed circuit is a loop of infinitely thin wire,^[16] and is invalid in other circumstances as discussed below. A different version, the Maxwell–Faraday equation (discussed below), is valid in all circumstances.

Quantitative

The definition of surface integral relies on splitting the surface Σ into small surface elements. Each element is associated with a vector dA of magnitude equal to the area of the element and with direction normal to the element and pointing “outward” (with respect to the orientation of the surface).

Faraday's law of induction makes use of the magnetic flux Φ_B 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:

$\Phi_B = \iint\limits_{\Sigma(t)} \mathbf{B}(\mathbf{r}, t) \cdot d \mathbf{A}\ ,$

where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field (also called "magnetic flux density"), and B·dA is a vector dot product (the infinitesimal amount of magnetic flux through the infinitesimal area element dA). 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, $\mathcal{E}$ , defined as the energy available from a unit charge that has travelled once around the wire loop.^[16]^[17]^[18]^[19] 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.

Faraday's law states that the EMF is also given by the rate of change of the magnetic flux:

$\mathcal{E} = -{{d\Phi_B} \over dt} \$ ,

where $\mathcal{E}$ is the electromotive force (EMF) and Φ_B is the magnetic flux. 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 Φ_B, Faraday's law of induction states that^[20]^[21]

$\mathcal{E} = -N {{d\Phi_B} \over dt}$

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

Maxwell–Faraday equation

An illustration of Kelvin-Stokes theorem with surface Σ its boundary ∂Σ and orientation n set by the right-hand rule.

The Maxwell–Faraday equation is a generalisation of Faraday's law that states that a time-varying magnetic field is always accompanied by a spatially-varying, non-conservative electric field, and vice versa. The Maxwell–Faraday equation is

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

(in SI units) where $\nabla\times$ is the curl operator and again E(r, t) is the electric field and B(r, t) is the magnetic field. These fields can generally be functions of position r and time t.

The Maxwell–Faraday equation is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It can also be written in an integral form by the Kelvin-Stokes theorem:^[22]

$\oint_{\partial \Sigma} \mathbf{E} \cdot d\boldsymbol{\ell} = - \int_{\Sigma} \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A}$

where, as indicated in the figure:

Σ is a surface bounded by the closed contour ∂Σ,
E is the electric field, B is the magnetic field.
dℓ is an infinitesimal vector element of the contour ∂Σ,
dA is an infinitesimal vector element of surface Σ. If its direction is orthogonal to that surface patch, the magnitude is the area of an infinitesimal patch of surface.

Both dℓ and dA have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin-Stokes theorem. For a planar surface Σ, a positive path element dℓ of curve ∂Σ is defined by the right-hand rule as one that points with the fingers of the right hand when the thumb points in the direction of the normal n to the surface Σ.

The integral around ∂Σ is called a path integral or line integral.

Notice that a nonzero path integral for E is different from the behavior of the electric field generated by charges. A charge-generated E-field can be expressed as the gradient of a scalar field that is a solution to Poisson's equation, and has a zero path integral. See gradient theorem.

The integral equation is true for any path ∂Σ through space, and any surface Σ for which that path is a boundary.

If the path Σ is not changing in time, the equation can be rewritten:

$\oint_{\partial \Sigma} \mathbf{E} \cdot d\boldsymbol{\ell} = - \frac{d}{dt} \int_{\Sigma} \mathbf{B} \cdot d\mathbf{A}.$

The surface integral at the right-hand side is the explicit expression for the magnetic flux Φ_B through Σ.

Einstein's view

Reflection on this apparent dichotomy was one of the principal paths that led Einstein to develop special relativity:
It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor.

The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated.

But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

Examples of this sort, together with unsuccessful attempts to discover any motion of the earth relative to the "light medium," suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest.
— Albert Einstein, On the Electrodynamics of Moving Bodies[27