Biot-Savart Law

Oersted's experiment (Section 4.2) established, qualitatively, that a current-carrying

conductor produces a magnetic field circling around it. The Biot-Savart law, formulated shortly

afterward by Jean-Baptiste Biot and Felix Savart from careful experimental measurement, gives this

same relationship in a precise, quantitative, vector form -- for the field due to an infinitesimally

small ELEMENT of current, from which the field of any full, extended current distribution can then

be built up by integration.

Statement of the law. Consider a small current element $I\,d\vec{l}$ -- a short length $d\vec{l}$

of a wire carrying current $I$, with $d\vec{l}$ pointing in the direction of current flow -- and a

field point $P$ located at position vector $\vec{r}$ from the element, with $\hat{r}$ the unit vector

along $\vec{r}$ and $\theta$ the angle between $d\vec{l}$ and $\hat{r}$. The Biot-Savart law states

that the small magnetic field $d\vec{B}$ produced at $P$ by this element is

or, in magnitude,

Here $\mu_0$ is the permeability of free space, a constant fixed by the definition of the SI

ampere at $\mu_0 = 4\pi\times 10^{-7}\ \text{T}\,\text{m/A}$, so that $\mu_0/4\pi = 10^{-7} \ \text{T}\,\text{m/A}$ exactly -- a combination that appears so often through this chapter that it

is worth memorising as a single number.

Direction: the right-hand (cross-product) rule. Because $d\vec{B}$ is given by a vector cross

product $d\vec{l}\times\hat{r}$, it is always perpendicular to the plane containing both $d\vec{l}$

and $\hat{r}$. Its precise direction is found by curling the fingers of the right hand from the

direction of $d\vec{l}$ toward the direction of $\hat{r}$ (through the smaller angle $\theta$

between them); the thumb then points along $d\vec{B}$. When the current element and the field point

both lie in the plane of the page, this typically means $d\vec{B}$ points either straight out of the

page (marked, by convention, with a dot) or straight into the page (marked with a cross).

Features of the law, and its analogy with Coulomb's law. The Biot-Savart law is often compared

directly to Coulomb's law for a point charge, $dE = kq/r^2$, since both are inverse-square laws.

Three differences, however, are essential to keep straight. First, the SOURCE is different: Coulomb's

law needs only a point charge $q$, while the Biot-Savart law needs a moving charge, represented here

by a current element $I\,d\vec{l}$ -- an isolated current element, unlike an isolated point charge,

cannot physically exist on its own (current must flow in a closed circuit), so $d\vec{B}$ is always,

in practice, one small piece of a full integral around a complete circuit. Second, the ANGLE

dependence: Coulomb's field has no $\sin\theta$ factor at all (a point charge's field is the same in

every direction), whereas the Biot-Savart field is proportional to $\sin\theta$ and vanishes entirely …