Chapter 12. Sources of Magnetic Fields
12.1 The Biot-Savart Law
Learning Objectives
By the end of this section, you will be able to:
- Explain how to derive a magnetic field from an arbitrary current in a line segment
- Calculate magnetic field from the Biot-Savart law in specific geometries, such as a current in a line and a current in a circular arc
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 (Figure 12.2), the magnetic field [latex]d\stackrel{\to }{\textbf{B}}[/latex] due to an element [latex]d\stackrel{\to }{\textbf{l}}[/latex] of a current-carrying wire is given by
The constant [latex]{\mu }_{0}[/latex] is known as the permeability of free space and is exactly
in the SI system. The infinitesimal wire segment [latex]d\stackrel{\to }{\textbf{l}}[/latex] is in the same direction as the current I (assumed positive), r is the distance from [latex]d\stackrel{\to }{\textbf{l}}[/latex] to P and [latex]\hat{\textbf{r}}[/latex] is a unit vector that points from [latex]d\stackrel{\to }{\textbf{l}}[/latex] to P, as shown in the figure.
The direction of [latex]d\stackrel{\to }{\textbf{B}}[/latex] is determined by applying the right-hand rule to the vector product [latex]d\stackrel{\to }{\textbf{l}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\hat{\textbf{r}}.[/latex] The magnitude of [latex]d\stackrel{\to }{\textbf{B}}[/latex] is
where [latex]\theta[/latex] is the angle between [latex]d\stackrel{\to }{\textbf{l}}[/latex] and [latex]\hat{\textbf{r}}.[/latex] Notice that if [latex]\theta =0,[/latex] then [latex]d\stackrel{\to }{\textbf{B}}=\stackrel{\to }{\textbf{0}}.[/latex] The field produced by a current element [latex]Id\stackrel{\to }{\textbf{l}}[/latex] has no component parallel to [latex]d\stackrel{\to }{\textbf{l}}.[/latex]
The magnetic field due to a finite length of current-carrying wire is found by integrating Equation 12.3 along the wire, giving us the usual form of the Biot-Savart law.
Biot-Savart law
The magnetic field [latex]\stackrel{\to }{\textbf{B}}[/latex] due to an element [latex]d\stackrel{\to }{\textbf{l}}[/latex] of a current-carrying wire is given by
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 fol