Electromagnetic_induction

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Faraday\'s law of induction (or the law of electromagnetic induction) is a term which is (ambiguously) applied to two related, but different, laws in classical electrodynamics.See pages 301–3 of: Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X. 

In one version, which (for the purposes of this article) we will call the "EMF version", Faraday\'s Law states that the induced electromotive force in a closed loop of wire is directly proportional to the time rate of change of magnetic flux through the loop.

In the other formulation, which (for the purposes of this article) we will call the "Maxwell\'s equations version" (since it is best known as one of the four modern Maxwell\'s equations), Faraday\'s Law states that a changing magnetic field can create an electric field.

(Note: the terminology "EMF version" and "Maxwell\'s equations version" are non-standard, and not used outside this article.)

The "Maxwell\'s equations version" is essentially a special case of the "EMF version"---namely, the case where the loop of wire is stationary in time.

Contents 1 "EMF version" of Faraday\'s law 2 "Maxwell\'s equations version" of Faraday\'s law 3 Relation between two versions 4 Magnetic flow meter 5 History 6 Electrical generators 7 Electrical transformers 8 See also 9 References

"EMF version" of Faraday\'s law

Moving a permanent magnet near a conductor (such as a metal wire) produces a voltage in that conductor. The resulting voltage is proportional to the speed of movement: moving the magnet twice as fast produces twice the voltage.

For a tightly-wound coil of wire, composed of N loops with the same area, Faraday\'s law (EMF version) states that

$\backslash mathcal\{E\}\; =\; -\; N\{\{d\backslash Phi\_B\}\; \backslash over\; dt\}$

where

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

N is the number of turns of wire

Φ_B is the magnetic flux in webers through a single loop. The direction of the electromotive force (the negative sign in the above formula) was first given by Lenz\'s law.

"Maxwell\'s equations version" of Faraday\'s law

A changing magnetic field can create an electric field; this phenomenon is described by Faraday\'s law (Maxwell\'s equations version):

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

where:

$\backslash nabla\backslash times$ denotes curl

E is the electric field

B is the magnetic field

This is called the differential form of this version of Faraday\'s law; by Kelvin-Stokes theorem it can also be written in an integral form:

$\backslash oint\_C\; \backslash mathbf\{E\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; -\; \backslash \; \{\; d\; \backslash over\; dt\; \}\; \backslash int\_S\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{A\}$

where

S is a surface enclosed by a contour C

More generally, the relation between the rate of change of the magnetic flux through the surface

E is the electric field,

dl is an infinitesimal element of the contour C,

B is the magnetic field.

dA is the a vector, whose magnitude is the area of an infinitesimal patch of surface, and whose direction is orthogonal to that surface patch.

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.

(The integral at the right-hand side of this equation is the explicit expression for the magnetic flux through S.)

Relation between two versions

Starting with the "EMF version", we can look at the special case where the wire-coil is stationary. EMF is defined as work per unit charge traversing the circuit, and due to the Lorentz force law this work per charge is the integral of the electric field E around the loop. (The magnetic force does not contribute to the EMF, being perpendicular to the direction of the loop, and therefore doing no work.) After plugging this in, the result is the integral form of the "Maxwell\'s equations version" of Faraday\'s law. Note that the "electric force" term (qE) of the Lorentz force law plays a big role in the derivation, while the "magnetic force" term (qv×B) only plays a role insofar as it proves that the magnetic force does not contribute to the EMF.

Likewise, putting the argument in reverse, the "Maxwell\'s equations version" of Faraday\'s law, along with the Lorentz force law, gives the "EMF version" in the special case where the wire-coil is stationary.

A different special case of the "EMF version", namely the special case where B is unchanging in time, is equivalent to the Lorentz force law.[1] Here, it is the "magnetic force" term that plays the primary role, while the electric force and "Maxwell\'s equations version" of Faraday\'s law only play a role in the derivations insofar as they show that the electric force does not contribute to the EMF.

In fact, assuming any two of these three equations--the Lorentz force law, the "EMF version", and the "Maxwell\'s equations version"--the third can be derived.

Magnetic flow meter

Faraday\'s law is used for measuring the flow of electrically conductive liquids and slurries. Such instruments are called magnetic flow meters. The induced voltage U generated in the magnetic field B due to a conductive liquid moving at velocity v is thus given by:

$\backslash mathbf\; U=\; BLv$,

where L is the distance between electrodes in the magnetic flow meter.

History

Faraday\'s law, along with the other laws of electromagnetism, was later incorporated into Maxwell\'s equations, unifying all of electromagnetism. Faraday\'s law of induction is based on Michael Faraday\'s experiments in 1831. The effect was also discovered by Joseph Henry at about the same time, but Faraday published first.Ulaby, Fawwaz (2007). Fundamentals of applied electromagnetics, 5th Edition, Pearson:Prentice Hall, p. 255. ISBN 0-13-241326-4. Joseph Henry. Distinguished Members Gallery, National Academy of Sciences. Retrieved on 2006-11-30.

See Maxwell\'s original discussion of electromotive force, James Clerk Maxwell (1881). A treatise on electricity and magnetism v. 2. Oxford UK: Clarendon Press, Chapter III, §530, p. 178. ISBN 0486606376.  or one of today\'s textbook treatments. BB Laud (1987). Electromagnetics. New Delhi: New Age International, p. 151. ISBN 0852264992. 

Lenz\'s law, formulated by German phyco Heinrich Lenz in 1834, gives the direction of the induced electromotive force and current resulting from electromagnetic induction.

Electrical generators

Main article: electrical generator

The EMF generated by Faraday\'s law (EMF version) is called an induced electromotive force (or induced EMF), and is the phenomenon underlying electrical generators. When a permanent magnet is rotated around a conductor, or vice versa, an electromotive force is created. If the wire is connected through an electrical load, current will flow, and thus electrical energy is generated.

When the current is flowing through the wire loop, a magnetic field will be generated through Ampere\'s law. The electromagnet thus created will resist the motion of the permanent magnet around the wire loop. The energy required to keep the permanent magnet moving, despite this resistive force, is exactly equal to the electrical energy generated (plus energy wasted due to friction and other inefficiencies). This is why electrical generators can convert mechanical energy to electrical energy.

Although Faraday\'s law (EMF version) always describes the working of electrical generators, the detailed mechanism can differ in different cases. When the magnet is rotated around a stationary conductor, the changing magnetic field creates an electric field, as described by the "Maxwell\'s equations version" of Faraday\'s law, and that electric field pushes the charges through the wire. On the other hand, when the magnet is stationary and the conductor is rotated, the moving charges experience a magnetic force (as described by the Lorentz force law), and this magnetic force pushes the charges through the wire. (This case is also called motional EMF.)For more information on motional EMF, induced EMF, Faraday\'s law, and the Lorentz force, see pages 301–3 of: Griffiths, David J. (1999). Introduction to electrodynamics, 3rd Edition, Prentice Hall. ISBN 013805326X. 

Electrical transformers

Main article: transformer

The "induced EMF" caused by Faraday\'s law (either version) is also responsible for electrical transformers. When the electric current in a loop of wire changes, the changing current creates a changing magnetic field. A second wire in reach of this magnetic field will experience this change in magnetic field as a change in its coupled magnetic flux. Therefore, an electromotive force is set up in the second loop. If the two ends of this loop are connected through an electrical load, current will flow.

See also

• Michael Faraday
• Magnetic field
• Magnetic flux
• Ampère\'s law
• Lenz\'s law
• Lorentz force
• Stokes\' theorem
• Vector calculus
• Moving magnet and conductor problem
• Crosstalk, an electrical interference caused by this law.

References

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