Chapter 35
Ampere’s Law`
In Chapter 31 we introduced the Biot-Savart law, which gives the magnetic field produced by a short current
element, and allows us to find the magnetic field due to any arbitrary geometry of electric current. Another
equation that gives the magnetic field produced by an electric current isAmpere’s law, named, like the SI unit of electric current, for the French physicist Andr ́e-Marie Ampere. Given an electric currentI, imagine
drawing a closed curveCaround the current, so that the current passes through a surface bounded byC.
Now divide the curveCinto small segmentsl, and at each segment, measure the component magnetic field
that is parallel tol; we’ll call that magnetic fieldBk. Then Amp`ere’s law states that
X
BklD 0 I: (35.1)
In other words, when we add together the productsBklfor all the segmentslthat make up curveC,we
get 0 times the current passing through the surface bounded byC.
So what? The Biot-Savart law tells us the magnetic field produced by an arbitrary arrangement of electric
current; why do we need another law that tells us the same thing? Recall Gauss’s law from Chapter 16: it
allows us to compute the electric field due to an arbitrary distribution of charge, although we could do the
same thing with Coulomb’s law. The difference is that Gauss’s law allows us to compute the electric field for
symmetricalcharge distributions very easily—much more easily than using Coulomb’s law. In these cases,
Gauss’s law can save a great deal of work. But if we have an irregular distribution of charge, we may have
no choice but to rely on Coulomb’s law and compute the electric field “the hard way.”
The relationship between the Biot-Savart law and Ampere’s law is similar. Although the Biot-Savart law will always work, it can be difficult to use. In some cases where the distribution of current is highly symmetrical, Ampere’s law gives us a shortcut for finding the magnetic field that is much less work than
using the Biot-Savart law. For irregular arrangements of electric current, though, we may have no choice but
to “do it the hard way” and resort to the Biot-Savart law.
For example, let’s find an expression for the magnetic field due to a currentIin an infinitely long, straight
wire, at a perpendicular distancerfrom the wire. To use Amp`ere’s law, we imagine drawing a circle of radius
raround the wire, so that the plane of the circle is perpendicular to the wire and the wire passes through the
center of the circle. We already know that the magnetic field due to the wire is in the shape of concentric
circles around the wire, so when we divide the circle into a number of small segmentsl, we know the
magnetic fieldBwill already be parallel tolfor each segment. Therefore for an infinitely long, straight
wire,
X
BklDB
X
lD2rB: (35.2)Then by Amp`ere’s law,
2rBD 0 I; (35.3)