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Mathematics for Physics

25 Biot-Savart Law

A law that is used to compute the magnetic field of a current element, where a current element is a short segment of wire carrying current.

The Biot–Savart law for a line current. The magnetic field of a steady line current is the integral of the Biot–Savart contribution of each length element. This principle is used to compute \(\mathbf{B}\) for wires, loops, and solenoids of known shape.

The Biot–Savart law for a line current is

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

where

  • \(\mathbf{B}(\mathbf{r})\) is the magnetic field at the field point.
  • \(I\) is the current.
  • \(d\mathbf{l}'\) is a directed length element of the wire.
  • \(r\) is the distance from the current element to the field point.
  • \(\hat{\mathbf{r}}\) is the unit vector from the current element to the field point.
  • \(\mu_{0}\) is the permeability of free space.

The volume-current form. The same law for a volume current replaces \(I\,d\mathbf{l}'\) by \(\mathbf{J}\,d\tau'\). This principle is used to compute \(\mathbf{B}\) of a distributed current.

The Biot–Savart law for a volume current 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

  • \(\mathbf{J}\) is the volume current density.
  • \(d\tau'\) is the volume element.

The field of a long straight wire. The field of a long straight wire falls as the inverse of the perpendicular distance. This principle is used to recover the standard result for an infinite wire.

The magnetic field of a long straight wire is

\[ B = \dfrac{\mu_{0}I}{2\pi s} \]

where

  • \(B\) is the magnitude of the magnetic field.
  • \(I\) is the current.
  • \(s\) is the perpendicular distance from the wire.
  • \(\mu_{0}\) is the permeability of free space.