(Redirected from Ampere's law)

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

[itex]\oint_S \mathbf{B} \cdot d\mathbf{s} = \mu_0 I_{\mathrm{enc}} [itex]

where

[itex]\mathbf{B}[itex] is the magnetic field,

[itex]d\mathbf{s}[itex] is an infinitesimal element (differential) of the closed loop [itex]S[itex],

[itex]I_{\mathrm{enc}}[itex] is the current enclosed by the curve [itex]S[itex],

[itex]\mu_0[itex] is the permeability of free space,

[itex]\oint_S[itex] is the integral along the closed loop [itex]S[itex].

## Generalized Ampre's law

James Clerk Maxwell noticed a logical inconsistency when applying Ampre'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 Ampre's law which was incorporated into Maxwell's equations. The generalized formula is as follows:

[itex]\oint_S \mathbf{B} \cdot d\mathbf{s} = \mu_0 I_{\mathrm{enc}} + \frac{d \mathbf{\Phi_E}}{dt}[itex]

where

[itex]\mathbf{\Phi_E}[itex] is the flux of electric field through the surface.

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

[itex]\nabla\times\vec B = \mu_0 \vec J + \mu_0 \epsilon_0 \frac{\partial\vec E}{\partial t}[itex]

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

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