Ampère's law

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Electrostatics

Electromagnetism
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Electric charge
Coulomb's law
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Gauss's law
Electric potential
Magnetostatics
Ampere's law
Magnetic field
Magnetic moment
Electrodynamics
Electric current
Lorentz force law
Electromotive force
Electromagnetic induction
Faraday-Lenz law
Displacement current
Maxwell's equations
Electromagnetic field
Electromagnetic radiation
Electrical circuits
Electrical conduction
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Capacitance
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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 $\vec{B}$ to its source, the current density $\vec{J}$ :

$\oint_C \frac{\vec{B}}{\mu_0} \cdot \mathrm{d}\vec{l} = \int\!\!\!\!\int_S \vec{J} \cdot \mathrm{d}\vec{A} = I_{\mathrm{enc}}$

where

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

C

$\vec{B}$ is the magnetic flux density in teslas,

$\mathrm{d}\vec{l}$ 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{A} \!\$ is a differential vector element of surface area A, 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

$\mu_0 = 4 \pi \times 10^{-7}$ is the permeability of free space (in henries per meter),

Equivalently, the original equation in differential form is

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

where

$\vec{\nabla}$ is the vector differential 'Del'

$\times\,$ is the cross product operator

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

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

[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 current density (in amperes 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.