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Mathematics for Physics

24 Ampere’s Law

A relation that is used to relate the magnetic field around a closed loop to the current through the enclosed surface, where a closed loop is a path that returns to its starting point.

Ampère’s law in integral form. The line integral of \(\mathbf{B}\) around a closed path equals \(\mu_{0}\) times the enclosed current. This principle is used to compute \(\mathbf{B}\) when symmetry makes the field constant on the path.

Ampère’s law in integral form is

\[ \oint\mathbf{B}\cdot d\mathbf{l} = \mu_{0}I_{\mathrm{enc}} \]

where

  • \(\mathbf{B}\) is the magnetic field.
  • \(d\mathbf{l}\) is a displacement along the loop.
  • \(I_{\mathrm{enc}}\) is the current through any surface bounded by the loop.
  • \(\mu_{0}\) is the permeability of free space.

Ampère’s law in differential form. In differential form the curl of \(\mathbf{B}\) is proportional to the local current density. This principle is used to write the local magnetostatic Maxwell equation.

Ampère’s law in differential form is

\[ \nabla\times\mathbf{B} = \mu_{0}\mathbf{J} \]

where

  • \(\nabla\times\) is the curl.
  • \(\mathbf{B}\) is the magnetic field.
  • \(\mathbf{J}\) is the volume current density.
  • \(\mu_{0}\) is the permeability of free space.

The Ampère–Maxwell correction. In time-dependent problems the enclosed current is completed by the displacement current. Displacement current is the term \(\epsilon_{0}\dfrac{\partial\mathbf{E}}{\partial t}\) that restores charge conservation. This principle is used to apply Ampère’s law to charging capacitors and electromagnetic waves.

The Ampère–Maxwell law is

\[ \oint\mathbf{B}\cdot d\mathbf{l} = \mu_{0}I_{\mathrm{enc}} + \mu_{0}\epsilon_{0}\dfrac{d\Phi_{E}}{dt} \]

where

  • \(\Phi_{E}\) is the electric flux through the surface bounded by the loop.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(t\) is time.