c/An Amperian` surface su-
perimposed on the landscape.d/An electron jumps through a
hoop.e/An alternative Amperian`
surface.going into the page. We’re used to dealing with currents made by
many charged particles, but logically we can’t have some minimum
number that would qualify as a current. This is not a static current,
however, because the current at a given point in space is not staying
the same over time. If the particle is pointlike, then it takes zero
time to pass any particular location, and the current is then infinite
at that point in space. A moment later, when the particle is passing
by some other location, there will be an infinite current there, and
zero current in the previous location. If this single particle qualifies
as a current, then it should be surrounded by a curly magnetic field,
just like any other current.^11
This explanation is simple and reasonable, but how do we know
it’s correct? Well, it makes another prediction, which is that the
positively charged particle should be making an electric field as well.
Not only that, but if it’s headed for the back of your head, then
it’s getting closer and closer, so the electric field should be getting
stronger over time. But this is exactly what Maxwell’s equations
require. There is no currentIthroughpiercing the Amp`erian surface
shown in figure c, so Maxwell’s equation for ΓB becomesc^2 ΓB =
∂ΦE/∂t. The only reason for an electric field to change is if there are
charged particles making it, and those charged particles are moving.
When charged particles are moving, they make magnetic fields as
well.
Note that the above example is also sufficient to prove the pos-
itive sign of the∂ΦE/∂tterm in Maxwell’s equations, which is dif-
ferent from the negative sign of Faraday’s−∂ΦB/∂tterm.
The addition of the∂ΦE/∂tterm has an even deeper and more
important physical meaning. With the inclusion of this term, Max-
well’s equations can describe correctly the way in which disturbances
in the electric and magnetic fields ripple outwards at the speed of
light. Indeed, Maxwell was the first human to understand that light
was in fact an electromagnetic wave. Legend has it that it was on a
starry night that he first realized this implication of his equations.
He went for a walk with his wife, and told her she was the only other
person in the world who really knew what starlight was.
To see how the∂ΦE/∂tterm relates to electromagnetic waves,
let’s look at an example where we would get nonsense without it.
Figure d shows an electron that sits just on one side of an imagi-
nary Amp`erian surface, and then hops through it at some randomly
(^11) One way to prove this rigorously is that in a frame of reference where the
particle is at rest, it has an electric field that surrounds it on all sides. If
the particle has been moving with constant velocity for a long time, then this
is just an ordinary Coulomb’s-law field, extending off to very large distances,
since disturbances in the field ripple outward at the speed of light. In a frame
where the particle is moving, this pure electric field is experienced instead as a
combination of an electric field and a magnetic field, so the magnetic field must
exist throughout the same vast region of space.
Section 11.6 Maxwell’s equations 723