Ampère's law

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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 B to its source, the current density J:

<math>\oint_C \frac{\mathbf{B}}{\mu_0} \cdot d\mathbf{l} = \int\!\!\!\!\int_S \mathbf{J} \cdot d \mathbf{A} = I_{\mathrm{enc}} </math>

where

<math>\oint_C</math> is the closed line integral around contour (closed curve) <math>C</math>.

<math>\mathbf{B}</math> is the magnetic flux density in teslas,

<math>d\mathbf{l}</math> is an infinitesimal element (differential) of the contour <math>C</math>,

<math>\mathbf{J}</math> is the current density (in amperes per square meter) through the surface S enclosed by contour C

<math> d \mathbf{A} \!\ </math> is a differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S

<math>I_{\mathrm{enc}} \!\ </math> is the current enclosed by the curve <math>C</math>, or strictly, the current that penetrates surface <math>S</math>,

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

Equivalently, the original equation in differential form is

<math>\nabla \times \mathbf{H} = \mathbf{J} </math>

where

<math>\nabla</math> is the vector differential 'Del'

<math>\times\,</math> is the cross product operator

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

<math> \mathbf{B} \ = \ \mu \mathbf{H} </math>

[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 <math>S</math> passes between the plates of the capacitor, and not through any wires, then <math>\mathbf{J} = 0</math> even though <math>\oint_C \mathbf{H} \cdot d\mathbf{l}\ne 0</math>. 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:

<math>\oint_C \mathbf{H} \cdot d\mathbf{l} = \iint_S \mathbf{J} \cdot d \mathbf{A} +

{d \over dt} \iint_S \mathbf{D} \cdot d \mathbf{A}</math>

where in linear media

<math> \mathbf{D} \ = \ \varepsilon \mathbf{E}</math>

is the displacement current density (in amperes per square meter).

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

<math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}</math>

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.