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Mathematics for Physics

68 Magnetic Fields

An alteration of space that is used to distinguish a magnetostatic field of steady currents from a field induced by a changing electric field.

Gauss’s law for magnetism. A magnetostatic field is divergenceless. This principle is used to write \(\mathbf{B}\) as the curl of a vector potential.

Gauss’s law for magnetism is

\[ \nabla\cdot\mathbf{B} = 0 \]

where

  • \(\nabla\cdot\) is the divergence.
  • \(\mathbf{B}\) is the magnetic field.

Ampère’s law. The curl of a magnetostatic field is proportional to the local current density. This principle is used to relate \(\mathbf{B}\) to its steady sources.

Ampère’s law 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. A changing electric field induces an additional magnetic field. This principle is used to complete Ampère’s law in time-dependent problems.

The Ampère–Maxwell law is

\[ \nabla\times\mathbf{B} = \mu_{0}\mathbf{J} + \mu_{0}\epsilon_{0}\dfrac{\partial\mathbf{E}}{\partial t} \]

where

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