12.1 The biot-savart law

We have seen that mass produces a gravitational field and also interacts with that field. Charge produces an electric field and also interacts with that field. Since moving charge (that is, current) interacts with a magnetic field, we might expect that it also creates that field—and it does.

The equation used to calculate the magnetic field produced by a current is known as the Biot-Savart law. It is an empirical law named in honor of two scientists who investigated the interaction between a straight, current-carrying wire and a permanent magnet. This law enables us to calculate the magnitude and direction of the magnetic field produced by a current in a wire. The Biot-Savart law    states that at any point P ( [link] ), the magnetic field $d\overrightarrow{B}$ due to an element $d\overrightarrow{l}$ of a current-carrying wire is given by

The constant ${\mu }_0$ is known as the permeability of free space    and is exactly

in the SI system. The infinitesimal wire segment $d\overrightarrow{l}$ is in the same direction as the current I (assumed positive), r is the distance from $d\overrightarrow{l}$ to P and $\widehat{r}$ is a unit vector that points from $d\overrightarrow{l}$ to P , as shown in the figure.

The direction of $d\overrightarrow{B}$ is determined by applying the right-hand rule to the vector product $d\overrightarrow{l}\times \widehat{r}.$ The magnitude of $d\overrightarrow{B}$ is

where $\theta$ is the angle between $d\overrightarrow{l}$ and $\widehat{r}.$ Notice that if $\theta =0,$ then $d\overrightarrow{B}=\overrightarrow{0}.$ The field produced by a current element $Id\overrightarrow{l}$ has no component parallel to $d\overrightarrow{l}.$

The magnetic field due to a finite length of current-carrying wire is found by integrating [link] along the wire, giving us the usual form of the Biot-Savart law.

Since this is a vector integral, contributions from different current elements may not point in the same direction. Consequently, the integral is often difficult to evaluate, even for fairly simple geometries. The following strategy may be helpful.