242 Week 7: Sources of the Magnetic Field
which is true foranycurved pathCthat goes around the infinitely long straight wire precisely one
time, so that the integral ofdθis eventually 2π(with therin the length element alongB~always
cancelling the 1/rdependence ofB~).
Note two things. One is that current carrying wires that donotpass through the closed loopC
do not contribute to the loop integral – the integral ofdθaround such a loop always adds up to zero
because it doesn’t go, and stay, around. If there were many wiresand not just one, we could use
superposition and show that this equation would still be true as long as we only add up the total
currentIthatpasses throughCon the right hand side.
The other is something that I can’t precisely show, but which you cankindof see is true. It turns
out that this equation workseven if the wire(s) aren’t infinitely long or straight! In fact it works
forany steady-state (static) current passing through the closed loopC. We can imagine trying to
prove this by (for example) leavingCas a circle and starting to deform the path followed by the
currentIand noting how the result depends on the angle subtended by each point on the circle, but
in the end visualizing it would be too difficult because of the cross product. For that reason, this is
one of the few times in this book where I’ll ask you to just trust me because I can’tquiteshow you
how the result doesn’t change as we, for example, bend the infinitelylong straight wire around into
an arbitrary loop itself, while maintaining the fact that it passes “through”^68.
This leads us to write the previous equation in the following carefully selected form:
∮CB~·d~ℓ=μ 0 Ithrough C=μ 0∫
S/CJ~·nˆdA (536)which we will call (the incorrect form of)Ampere’s Law. Ampere’s Law is our third Maxwell
equation, and is the equation that Maxwell in fact “fixed” to get his name on the entire set. Some
fix!
Note that I wrote the current “throughC” in a mathematically correct way as theflux of the
current density through an open surfaceSbounded by the closed curveC!^69. This is the way I’d
like you to practice writing Ampere’s Law, although inapplicationbelow we’ll often just add up the
total current through any given loop by inspection.
Also note thatμ 0 = 4πkm– a form that is rather reminiscent of the 4πkewe saw in Gauss’s
Law for Electricity. Keep that loosely in mind for later, as eventually itwill help us do a simple but
enormously important piece of arithmetic without a calculator.
Another Maxwell Equation! Gosh, seems as though it should be goodforsomething, doesn’t it?
Indeed it is. Even in its slightly broken form above, we can use it tofind the magnetic field in
problems with just enough of the right kind of symmetry. There areonly a handful of problems that
fit the bill, but they are all useful and important. In the end, though, the purpose of Ampere’s Law
(fixed) is that it is a law of nature (where the Biot-Savart Law per seis not) and in fact we usually
derive the Biot-Savart Law fromitin modern times, or better yet skip directly to relativistically
correct, retarded descriptions of the electromagnetic field thatarereallydifficult to evaluate (but
correct).
For now, though, and in this course, we’ll content ourselves withthehandful of problems where
Ampere’s Law can be used to evalute the magnetic field. We’re basicallygoing to do all of them.
(^68) This is your first hint. Exactly what does it mean for a currentto pass “through” anarbitraryclosed loop? It’s
easy to answer this when the line is straight and the loop is a nice plane circle, but Ampere’s Law holds for curves
Cthat are topologically equivalent to a kilometer of fishing line with the ends tied together (to form a closed curve)
and the balled up onto the biggest, worst fishing tangle you ever saw! What does it mean for current to go “through”
that? And yet it can, and if you stuff the entire snarl into a pipe carrying water, you can completely imagine that
some of the water does, in fact, gothrough the loopas it flows along.
(^69) Hint: There aremany– in fact, an infinite number of – surfacesSthat are bounded byany particularclosed
curveC. Is the value of this integral independent of which surface you choose? If it is, is that a problem?