f/A magnetic field in the form of
a sine wave.
g/The wave pattern is curly.
For example, the circulation
around this reactangle is nonzero
and counterclockwise.
chosen moment. Unadorned with the∂ΦE/∂tterm, Maxwell’s equa-
tion for ΓBreads asc^2 ΓB= 4πkIthrough, which is Amp`ere’s law. If
the electron is a pointlike particle, then we have an infinite current
Ithroughat the moment when it pierces the imaginary surface, and
zero current at all other times. An infinite magnetic circulation ΓB
can only be produced by an infinite magnetic field, so without the
∂ΦE/∂tterm, Maxwell’s equations predict nonsense: the edge of the
surface would experience an infinite magnetic field at one instant,
and zero magnetic field at all other times. Even if the infinity didn’t
upset us, it doesn’t make sense that anything special would happen
at the moment the electron passed through the surface, because the
surface is an imaginary mathematical construct. We could just as
well have chosen the curved surface shown in figure e, which the
electron never crosses at all. We are already clearly getting non-
sensical results by omitting the∂ΦE/∂tterm, and this shouldn’t
surprise us because Amp`ere’s law only applies to statics. More to
the point, Amp`ere’s law doesn’t have time in it, so it predicts that
this effect is instantaneous. According to Amp`ere’s law, we could
send Morse code signals by wiggling the electron back and forth, and
these signals would be received at distant locations instantly, with-
out any time delay at all. This contradicts the theory of relativity,
one of whose predictions is that information cannot be transmitted
at speeds greater than the speed of light.
Discussion Questions
A Induced magnetic fields were introduced in the text via the imaginary
landscape shown in figure b on page 722, and I argued that the magnetic
field could have been produced by a positive charge coming from behind
your head. This is a specific assumption about thenumberof charges
(one), thedirectionof motion, and thesignof the charge. What are some
other scenarios that could explain this field?11.6.2 Light waves
We could indeed send signals using this scheme, and the signals
wouldbea form of light. A radio transmitting antenna, for instance,
is simply a device for whipping electrons back and forth at mega-
hertz frequencies. Radio waves are just like visible light, but with a
lower frequency. With the addition of the∂ΦE/∂tterm, Maxwell’s
equations are capable of describing electromagnetic waves. It would
be possible to use Maxwell’s equations to calculate the pattern of the
electric and magnetic fields rippling outward from a single electron
that fidgets at irregular intervals, but let’s pick a simpler example
to analyze.
The simplest wave pattern is a sine wave like the one shown
in figure f. Let’s assume a magnetic field of this form, and see
what Maxwell’s equations tell us about it. If the wave is traveling
through empty space, then there are no charges or currents present,724 Chapter 11 Electromagnetism