## The Biot-Savart Law

8.1 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

### Biot-Savart Law

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*, the magnetic field due to an element of a current-carrying wire is given by

### Long, Straight Wire

Recall that magnetic field lines are always closed loops. We saw this in the video at the beginning of the module. If you take a long wire and hold it perfectly straight, then run a current through it, it will produce a magnetic field that circulates around the wire.

Depending on the direction of the current, the B-field will circulate either clockwise or counterclockwise around the wire. We have another Right Hand Rule that helps us figure out the direction of the B-field produced by a current. In this RHR, grab the wire with your right hand so that your thumb points along the wire in the direction of the current.

Then your fingers will curl around the wire in the same direction that the B-field lines go.

To calculate the strength of the magnetic field produced by a current in a long straight wire, we use

where *I* is the current in the wire, *r* is the distance from the wire to where you are measuring the field, and µ_{0} is a constant, called the permeability of free space. This is kind of like the magnetic version of ε_{0} that you saw when we were talking about electric fields. (We can derive this formula using the Biot-Savart Law, though we will not discuss this derivation in this class.)

When looking at this formula for the magnetic field produced by a long straight wire, you can see that *B* is proportional to *I* so a stronger current will produce a stronger B-field, and it’s inversely proportional to *r*, so *B* decreases as you get farther away from the wire.

**Practice!**

Practice 8.1.1 |
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What is the direction of the magnetic field at point P, which is exactly in the middle of two parallel wires carrying equal currents I in opposite directions? |

Practice 8.1.2 |
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If the currents in these wires have the same magnitude but opposite directions, what is the direction of the magnetic field at point P? |

Practice 8.1.3 |
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Each of the wires in the figures below carry the same current, either into or out of the page. In which case is the magnetic field at the center of the square greatest? |

### Circular Wire Loop

If you take a wire and wrap it around so it forms a circular loop, you can find the strength of the magnetic field at the center of the loop with

where *I* is the current in the wire loop and *R* is the radius of the loop. You can use the same RHR I described earlier to find the direction of the B-field produced at the center of the loop. Here is a picture to show you how:

**Practice!**

Practice 8.1.4 |
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What is the direction of the magnetic field at point P, which is at the center of a semicircular loop of wire carrying a current I as shown? |

Practice 8.1.5 |
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What is the magnetic field in the center of the circular current loop? |