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Illustration 28.3: Ampere's Law and Symmetry
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A single wire carrying current in the z direction (out of the computer screen) has radial symmetry about the center of the wire. Two systems that differ only by a rotation about the center of the wire will be indistinguishable. This symmetry is, however, broken if a second wire is added, because the displacement vector from the first to the second wire defines a unique direction. Calculations to determine magnetic field strength that depend on the ability to follow a closed Amperian path are much more difficult because it is not possible to write a simple analytic expression for a path along which |B| is constant.
Look at the magnetic field vectors for the one-wire configuration. Notice how there is a circular symmetry about the center of the wire. Because of this symmetry, we can use Ampere's law to determine the magnetic field. Now look at the configuration with two wires. You can drag the wires either toward or away from each other. Notice that with two wires the magnetic field lines no longer have a circular symmetry. As a consequence we cannot use Ampere's law to determine the magnetic field. Do not think that Ampere's law is no longer valid. Ampere's law is always true. It is just that in certain cases it is easy to use Ampere's law to calculate the magnetic field and in others it is too difficult.
What is the analytic expression for the magnetic field on a path that has constant |B| if there is only one wire? Move the wires closer together and farther apart. Under what circumstances can this expression be used as an approximation in the case of two wires? Around one wire, |B| = μ0 I / 2 π r and it points in the direction tangent to a circle centered on the wire. If there are two wires, we may add together the magnetic fields due to each individual wire. Be careful: You must add these fields as vectors, not as just numbers.
The magnetic field produced by two long, straight wires is far from irregular. What types of symmetry does this system still have? There is still a symmetry in the z direction, but this symmetry is not useful for calculations using Ampere's law. Why? How do you use an Amperian loop to calculate the magnetic field? To use this symmetry, the loop or rectangle would have to be centered on the wire and have one side along the z axis and the other side in the xy plane. Using this rectangle, how much current is enclosed? Since the loop is infinitesimally thin, there is no current enclosed, and therefore the result for the magnetic field is zero. We already know that the magnetic field in the z direction and the radial direction are zero. So this symmetry also tells us something about the field.