21.8

Ampère’s Law

CONCEPTS AT A GLANCE We have seen that an electric current creates a magnetic field. However, the magnitude and direction of the field at any point in space depends on the specific geometry of the wire carrying the current. For instance, distinctly different magnetic fields surround a long, straight wire, a circular loop of wire, and a solenoid. Although different, each of these fields can be obtained from a general law known as Ampère’s law, which is valid for a wire of any geometrical shape. The Concepts-at-a-Glance chart in Figure 21-38 emphasizes that the concepts of electric current and magnetic field play the central roles in Ampère’s law, which specifies the relationship between an electric current and the magnetic field that it creates.

Figure 21-38 CONCEPTS AT A GLANCE Electric currents produce magnetic fields, and the relation between the two is expressed by Ampère’s law. The photograph shows a technician winding a superconducting wire onto a coil for use as an electromagnet. (Maximilian Stock, Ltd./Science Photo Library/Photo Researchers)

To see how Ampère’s law is stated, consider Figure 21-39, which shows two wires carrying currents I1 and I2. In general, there may be any number of currents. Around the wires we construct an arbitrarily shaped but closed path. This path encloses a surface and is constructed from a large number of short segments, each of length . Ampère’s law deals with the product of and for each segment, where is the component of the magnetic field that is parallel to (see the blow-up view in the drawing). For magnetic fields that do not change as time passes, the law states that the sum of all the terms is proportional to the net current I passing through the surface enclosed by the path. For the specific example in Figure 21-39, we see that I = I1 + I2. Ampère’s law is stated in equation form as follows:

| Ampère’s Law for Static Magnetic Fields
| For any current geometry that produces a magnetic field that does not change in time,

| (21.8)

where is a small segment of length along a closed path of arbitrary shape around the current, is the component of the magnetic field parallel to , I is the net current passing through the surface bounded by the path, and is the permeability of free space. The symbol indicates that the sum of all terms must be taken around the closed path.

Figure 21-39 This setup is used in the text to explain Ampère’s law.

To illustrate the use of Ampère’s law, we apply it in Example 11 to the special case of the current in a long, straight wire and show that it leads to the proper expression for the magnetic field.

| Example 11 | | | An Infinitely Long, Straight, Current-Carrying Wire

Use Ampère’s law to obtain the magnetic field produced by the current in an infinitely long, straight wire. Reasoning Figure 21-25a shows that compass needles point in a circular pattern around the wire, so we know that the magnetic field lines are circular. Therefore, it is convenient to use a circular path of radius r when applying Ampère’s law, as Figure 21-40 indicates.

Figure 21-40 Example 11 uses Ampère’s law to find the magnetic field in the vicinity of this long, straight, current-carrying wire.

Solution Along the circular path in Figure 21-40, the magnetic field is everywhere parallel to and has a constant magnitude, since each point is at the same distance from the wire. Thus, and, according to Ampère’s law, we have

But is just the circumference of the circle, so Ampère’s law reduces to

Dividing both sides by shows that , as given earlier in Equation 21.5.

Test your understanding of the material in Sections 21.5, 21.6, 21.7 and 21.8 :

· | The Force on a Current in a Magnetic Field
· | The Torque on a Current-Carrying Coil
· | Ampère’s Law
· | Magnetic Fields Produced by Currents

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.