In physics, Ampère's law is the magnetic equivalent of Gauss's law, discovered by André-Marie Ampère. 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 Ampère's law

James Clerk Maxwell noticed a logical inconsistency when applying Ampère'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 Ampère'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 Ampère-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 Ampère's law.

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