Magnetic Field due to a Current: Biot-Savart Law

Sections 10.1 through 10.9 dealt entirely with the FIRST half of this chapter's subject: given a magnetic field, what force (and torque) does it exert on a moving charge or a current? The remaining sections turn to the SECOND half: given a current, what magnetic field does IT produce? The qualitative answer -- a current-carrying wire is surrounded by a magnetic field circling around it, in the sense given by the right-hand thumb rule -- was already stated back in Section 10.1. What has been missing so far is a way to calculate this field QUANTITATIVELY, for a current of any given strength and a wire of any given shape, at any point in space.

That quantitative tool is the Biot-Savart law, an experimentally-deduced result (much as Coulomb's law was, in electrostatics) for the differential magnetic field $d\vec{B}$ produced at a point P by a small current element $Id\vec{l}$ (Fig. 10.15). If $\vec{r}$ is the position vector from the current element to the point P (of magnitude $r$), and $\theta$ is the angle between $d\vec{l}$ and $\vec{r}$, the MAGNITUDE of the differential field is

where $\mu_0=4\pi\times10^{-7}$ T.m/A is a fundamental constant called the permeability of free space (or the "permeability constant"), fixing the overall strength of magnetic effects in vacuum, exactly as $\varepsilon_0$ (or the constant $1/4\pi\varepsilon_0$) fixes the strength of electrostatic effects in Coulomb's law. Notice that this is, at its core, still an INVERSE-SQUARE law -- the field falls off as $1/r^2$ -- exactly the same power-law dependence as Coulomb's law for a point charge, even though the "source" here is a moving-charge current element rather than a static point charge.

The DIRECTION of $d\vec{B}$ is fixed by the vector cross product $d\vec{l}\times\hat{r}$ (where $\hat{r}$ is the unit vector along $\vec{r}$), giving the full vector form of the Biot-Savart law:

$d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l}\times\hat{r}}{r^2} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l}\times\vec{r}}{r^3}$ …