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Faraday's law of induction - Wikipedia, the free encyclopedia

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Faraday's law of induction

From Wikipedia, the free encyclopedia

Faraday's law of induction gives the relation between the rate of change of the magnetic flux through the surface S enclosed by a contour C and the electric field along the contour:

where E is the electric field, dl is an infinitesimal element of the contour C and B is the magnetic flux density. The are assumed to be related by the right-hand rule. directions of the contour C and of Equivalently, the differential form of Faraday's law is

which is one of the Maxwell equations. In the case of an inductor coil where the electric wire makes N turns, the formula becomes:

where V is the induced electromotive force and d/dt is the time-rate of change of magnetic flux . The direction of the electromotive force (the negative sign in the above formula) was first given by Lenz's law. 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.

See also

induction magnetic flux density Maxwell's equations Michael Faraday Ampere's law Stokes' theorem Vector calculus Retrieved from "http://en.wikipedia.org/wiki/Faraday%27s_law_of_induction" Categories: Electrodynamics | Introductory physics | Eponymous laws

This page was last modified 13:33, 29 March 2006. All text is available under the terms of the GNU Free Documentation License (see Copyrights for details).

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Faraday's Law

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Faraday's Law

Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc.

Index Faraday's Law concepts

Further comments on these examples Faraday's law is a fundamental relationship which comes from Maxwell's equations. It serves as a succinct summary of the ways a voltage (or emf) may be generated by a changing magnetic environment. The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil. It involves the interaction of charge with magnetic field.

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Faraday's Law

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Lenz's law AC coil example Faraday's Law and Auto Ignition HyperPhysics*****Electricity and magnetism

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Lenz's Law

When an emf is generated by a change in magnetic flux according to Faraday's Law, the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the examples below, if the B field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant.

Index Faraday's Law concepts

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Faraday's Law

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Magnet and Coil

When a magnet is moved into a coil of wire, changing the magnetic field and magnetic flux through the coil, a voltage will be generated in the coil according to Faraday's Law. In the example shown below, when the magnet is moved into the coil the galvanometer deflects to the left in response to the increasing field. When the magnet is pulled back out, the galvanometer deflects to the right in response to the decreasing field. The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change that produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. This inherent behavior of generated magnetic fields is summarized in Lenz's Law.

Index Faraday's Law concepts

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Faraday's Law

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Stokes' theorem - Wikipedia, the free encyclopedia

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Stokes' theorem

From Wikipedia, the free encyclopedia

Stokes' theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. It is named after Sir George Gabriel Stokes (1819-1903), although the first known statement of the theorem is by William Thomson (Lord Kelvin) and appears in a letter of his to Stokes. The theorem acquired its name from Stokes' habit of including it in the Cambridge prize examinations. Let M be an oriented piecewise smooth manifold of dimension n and let be an n-1 form that is a compactly supported differential form on M of class C1. If M denotes the boundary of M with its induced orientation, then

Here d is the exterior derivative, which is defined using the manifold structure only. The Stokes theorem can be considered as a generalisation of the fundamental theorem of calculus; and the latter indeed follows easily from the former. The theorem is often used in situations where M is an embedded oriented submanifold of some bigger manifold on which the form is defined. The theorem easily extends to linear combinations of piecewise smooth submanifolds, so-called chains. The Stokes theorem then shows that closed forms defined up to an exact form can be integrated over chains defined only up to a boundary. This is the basis for the pairing between homology groups and de Rham cohomology. The classical Kelvin-Stokes theorem:

which relates the surface integral of the curl of a vector field over a surface in Euclidean 3 space to the line integral of the vector field over its boundary, is a special case of the general Stokes theorem (with n = 2) once we identify a vector field with a 1 form using the metric on Euclidean 3 space. It can be rewritten for the student unacquainted with forms as

where P, Q and R are the components of F. These variants are frequently used:

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Stokes' theorem - Wikipedia, the free encyclopedia

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Likewise the Ostrogradsky-Gauss theorem (also known as the Divergence theorem or Gauss' theorem)

is a special case if we identify a vector field with the n-1 form obtained by contracting the vector field with the Euclidean volume form. The fundamental theorem of calculus and Green's theorem are also special cases of the general Stokes theorem. The general form of the Stokes theorem using differential forms is more powerful than the special cases, of course, although the latter are more accessible and are often considered more convenient by practicing scientists and engineers.

References

Stewart, James. Calculus: Concepts and Contexts. 2nd ed. Pacific Grove, CA: Brooks/Cole, 2001. Retrieved from "http://en.wikipedia.org/wiki/Stokes%27_theorem" Categories: Differential topology | Differential forms | Vector calculus | Duality theories | Mathematical theorems

This page was last modified 20:24, 20 March 2006. All text is available under the terms of the GNU Free Documentation License (see Copyrights for details). Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. Privacy policy About Wikipedia Disclaimers

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