Ampere's law - Definition


Ampere's law - Definition

In physics, Amp�re's law is the magnetic equivalent of Gauss's law, discovered by Andr�-Marie Amp�re. It relates the circulating magnetic field in a closed loop to the electric current passing through the loop:

<math>\oint_S \mathbf{B} \cdot d\mathbf{s} = \mu_0 I_{\mathrm{enc}} <math>

where

<math>\mathbf{B}<math> is the magnetic field,

<math>d\mathbf{s}<math> is an infinitesimal element (differential) of the closed loop <math>S<math>,

<math>I_{\mathrm{enc}}<math> is the current enclosed by the curve <math>S<math>,

<math>\mu_0<math> is the permeability of free space,

<math>\oint_S<math> is the path integral along the closed loop <math>S<math>.

James Clerk Maxwell noticed a logical inconsistency when applying Amp�re's law on charging capacitors, and thus 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 Amp�re's law which was incorporated into Maxwell's equations. The generalized formula is as follows:

<math>\oint_S \mathbf{B} \cdot d\mathbf{s} = \mu_0 I_{\mathrm{enc}} + \frac{d \mathbf{\Phi_E}}{dt}<math>

where

<math>\mathbf{\Phi_E}<math> is the flux of electric field through the surface.

This Amp�re-Maxwell law can also be stated in differential form:

<math>\nabla\times\vec B = \mu_0 \vec J + \mu_0 \epsilon_0 \frac{\partial\vec E}{\partial t}<math>

where the second term arises from the displacement current; omitting it yields the differential form of the original Amp�re's law.

See also