# 12.1 The biot-savart law

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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 ( [link] ), the magnetic field $d\stackrel{\to }{B}$ due to an element $d\stackrel{\to }{l}$ of a current-carrying wire is given by

$d\stackrel{\to }{B}=\frac{{\mu }_{0}}{4\pi }\phantom{\rule{0.2em}{0ex}}\frac{Id\stackrel{\to }{l}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{^}{r}}{{r}^{2}}.$

The constant ${\mu }_{0}$ is known as the permeability of free space    and is exactly

${\mu }_{0}=4\pi \phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{\text{−7}}\text{T}\cdot \text{m/A}$

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

The direction of $d\stackrel{\to }{B}$ is determined by applying the right-hand rule to the vector product $d\stackrel{\to }{l}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{^}{r}.$ The magnitude of $d\stackrel{\to }{B}$ is

$dB=\frac{{\mu }_{0}}{4\pi }\phantom{\rule{0.2em}{0ex}}\frac{I\phantom{\rule{0.2em}{0ex}}dl\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.1em}{0ex}}\theta }{{r}^{2}}$

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

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.

## Biot-savart law

The magnetic field $\stackrel{\to }{B}$ due to an element $d\stackrel{\to }{l}$ of a current-carrying wire is given by

$\stackrel{\to }{B}=\frac{{\mu }_{0}}{4\pi }\underset{\text{wire}}{\int }\frac{I\phantom{\rule{0.2em}{0ex}}d\stackrel{\to }{l}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{^}{r}}{{r}^{2}}.$

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.

## Problem-solving strategy: solving biot-savart problems

To solve Biot-Savart law problems, the following steps are helpful:

1. Identify that the Biot-Savart law is the chosen method to solve the given problem. If there is symmetry in the problem comparing $\stackrel{\to }{B}$ and $d\stackrel{\to }{l},$ Ampère’s law may be the preferred method to solve the question.
2. Draw the current element length $d\stackrel{\to }{l}$ and the unit vector $\stackrel{^}{r},$ noting that $d\stackrel{\to }{l}$ points in the direction of the current and $\stackrel{^}{r}$ points from the current element toward the point where the field is desired.
3. Calculate the cross product $d\stackrel{\to }{l}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{^}{r}.$ The resultant vector gives the direction of the magnetic field according to the Biot-Savart law.
4. Use [link] and substitute all given quantities into the expression to solve for the magnetic field. Note all variables that remain constant over the entire length of the wire may be factored out of the integration.
5. Use the right-hand rule to verify the direction of the magnetic field produced from the current or to write down the direction of the magnetic field if only the magnitude was solved for in the previous part.

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