240 Week 7: Sources of the Magnetic Field
7.5: Ampere’s Law
It is difficult to know the best way to show you the path from the Biot-Savart Law to Ampere’s
Law. In a sense, this is conceptually one of the most difficult things tosee, because it requires math
that you almost certainly haven’t seen yet. There is also little point in pursuing the formally correct
“best” path – as we noted above, the Biot-Savart Law is itself not strictly correct outside of the
context of magnetostatics where Ampere’s Law – with Maxwell’s eventual addition – is one of the
two equations that lead us toelectrodynamics– the fully unified dynamic electromagnetic field.
In fact, my favorite way of presenting this whole chapter is as a sort of detective story, where I
lay down hints along the way and give you a chance to win a prize^66. The goal is to see theflawin
Ampere’s Law as we soon write it, to see how itmustfail to be mathematically consistent for certain
geometries of currents, and – naturally – to correctly derive thefixfor it: the Maxwell Displacement
Current that (with Faraday’s Law) unified Electricity and Magnetism.
Of course, any student who wishes can skip ahead a few chapters and “cheat”, but that would
be no more satisfying than reading the last chapter of a mystery novel first.
So come on. You’re pretty smart. You’re taking a no-kidding physicscourse. Think you can
slam-dunk like James Clerk Maxwell? Thenbring it. Figure out why Ampere’s law isn’t consistent
and make it right, without peeking! If you can do that, you can do anything, and the knowledge
that you can do anything is more valuable than you can imagine when later you hit some really
difficult problems in life if not physics.
Thus we will start our Stoke’s Theorem Free discussion^67 by thinking once again about a single,
infinitely long, straight line of currentI. We recall from our Biot-Savart derivation above that:
B=^2 kmI
r=μ^0 I
2 πr(531)
where the direction of the magnetic field is around the currentIin the right-handed sense. The
geometry of this is drawn in figure 84.
Ir BFigure 84: An infinitely long straight wire carries currentIand has a magnetic field that goesaround
the wire in circular loops of constant magnitude in the right-handed sense.
Note well that the fielddrops off like 1 /r. The circumference of the circle just happens toincrease
liker. In week 3 we saw that if we multipled a field that went down like 1/r^2 by an area that went
(^66) A prize of no value whatsoever and of the greatest value you can imagine. In my own class, I up the ante of the
“no value whatsoever” part and give any student that managesit a piece of candy or a small prize picked out of a
treasure chest of cheap prizes. This, of course, makes the total value of the prizegreaterthan the greatest value you
can imagine, if you can imagine that. Of course, you can’t...
(^67) That, and the curl, are the math that we would be using if we were going to do this “right”.