Ampère's law

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Electrostatics

Electromagnetism
Electricity · Magnetism
Electric charge
Coulomb's law
Electric field
Gauss's law
Electric potential
Electric dipole moment
Magnetostatics
Ampère's law
Magnetic field
Magnetic dipole moment
Electrodynamics
Electric current
Lorentz force law
Electromotive force
(EM) Electromagnetic induction
Faraday-Lenz law
Displacement current
Maxwell's equations
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(EM) Electromagnetic radiation
Electrical Network
Electrical conduction
Electrical resistance
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An electric current produces a magnetic field.

In physics, Ampère's law, discovered by André-Marie Ampère, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. It is the magnetic equivalent of Faraday's law of induction.

1 Original Ampère's law
2 Corrected Ampère's law: the Ampère-Maxwell equation
3 See also
4 References
5 External links

[edit] Original Ampère's law

In its original form, Ampère's law relates the magnetic field $\vec{H}$ to its source, the current density $\vec{J}$ :

$\oint_C \vec{H} \cdot \mathrm{d}\vec{l} = \int\!\!\!\!\int_S \vec{J} \cdot \mathrm{d}\vec{S} = I_{\mathrm{enc}}$

where

$\oint_C$ is the closed line integral around contour (closed curve) C.

$\vec{H}$ is the magnetic field in amperes per metre.

$\mathrm{d}\vec{l}$ is an infinitesimal element (differential) of the contour C,

$\vec{J}$ is the current density (in amperes per square meter) through the surface S enclosed by contour C

$\mathrm{d}\vec{S} \!\$ is a differential vector area element of surface S, with infinitesimally small magnitude and direction normal to surface S,

$I_{\mathrm{enc}} \!\$ is the current enclosed by the curve C, or strictly, the current that penetrates surface S.

Equivalently, the original equation in differential form is

$\vec{\nabla} \times \vec{H} = \vec{J}$

where

$\vec{\nabla} \times \!\$ is the curl operator.

The magnetic field $\vec{H}$ in linear media, is related to the magnetic flux density $\vec{B}$ (in teslas) by

$\vec{B} \ = \ \mu \vec{H}$

where $\mu \!\$ is the permeability of the medium (in henries per meter), which by definition is $4 \pi \times 10^{-7}$ in free space. In non-linear media, $\mu \!\$ is a rank-2 tensor.

[edit] Corrected Ampère's law: the Ampère-Maxwell equation

James Clerk Maxwell noticed a logical inconsistency when applying Ampère's law to a charging or discharging capacitor. If surface $S$ passes between the plates of the capacitor, and not through any wires, then $\vec{J} = 0$ even though $\oint_C \vec{H} \cdot \mathrm{d}\vec{l}\ne 0$ . He concluded that this law had to be incomplete. To resolve the problem, he came up with the concept of displacement current and made a generalized version of Ampère's law which was incorporated into Maxwell's equations.

The generalized law, as corrected by Maxwell, takes the following integral form:

$\oint_C \vec{H} \cdot \mathrm{d}\vec{l} = \iint_S \vec{J} \cdot \mathrm{d} \vec{A} + {\mathrm{d} \over \mathrm{d}t} \iint_S \vec{D} \cdot \mathrm{d} \vec{A}$

where in linear media

$\vec{D} \ = \ \varepsilon \vec{E}$

is the displacement flux density (in coulombs per square meter).

This Ampère-Maxwell law can also be stated in differential form:

$\vec{\nabla} \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}$

where the second term arises from the displacement current.

With the addition of the displacement current, Maxwell was able to postulate (correctly) that light was a form of electromagnetic wave. See Electromagnetic wave equation for a discussion on this important discovery.

[edit] See also

[edit] References

Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 013805326X.
Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0716708108.