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Mathematics for Physics

67 Magnetic Field

A vector field produced by moving charges or currents that is used to give the magnetic force on other moving charges or currents, where a vector field is an assignment of a vector to each point in space.

The magnetic force that defines \(\mathbf{B}\). The magnetic force on a charge is perpendicular to both the velocity and the field. This principle is used to define \(\mathbf{B}\) from the force on a moving test charge.

The magnetic force is

\[ \mathbf{F}_{\mathrm{mag}} = q\bigl(\mathbf{v}\times\mathbf{B}\bigr) \]

where

  • \(\mathbf{F}_{\mathrm{mag}}\) is the magnetic force.
  • \(q\) is the charge.
  • \(\mathbf{v}\) is the velocity.
  • \(\mathbf{B}\) is the magnetic field.

The Biot–Savart field of a current. A steady current distribution produces a magnetic field given by the Biot–Savart law. This principle is used to map the magnetostatic field of a known current.

The Biot–Savart law is

\[ \mathbf{B}(\mathbf{r}) = \dfrac{\mu_{0}}{4\pi}\displaystyle\int\dfrac{\mathbf{J}(\mathbf{r}')\times\hat{\mathbf{r}}}{r^{2}}\,d\tau' \]

where

  • \(\mathbf{B}(\mathbf{r})\) is the magnetic field at the field point.
  • \(\mathbf{J}\) is the volume current density.
  • \(r\) is the distance from the source element to the field point.
  • \(\hat{\mathbf{r}}\) is the unit vector from the source element to the field point.
  • \(d\tau'\) is the volume element.
  • \(\mu_{0}\) is the permeability of free space.

The absence of magnetic monopoles. There are no magnetic monopoles, so the magnetic 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.