On 5/25/2011 1:12 AM, Mark Iverson wrote:
Just wanted to throw out a question to the Vort Collective...
In an EM wave, why are the electric and magnetic fields perpendicular
to each other?
The answer to the question is really quite simple and it comes from our
definition of what these fields are - which is in turn dictated by what
we can measure with instruments. The most fundamental quantity related
to these fields that nature seems to possess is a 3 dimensional time
varying charge displacement field whose dynamic characteristics are
excellently described by Maxwell's equations and whose definition seems
most completely given by the two components which are conventionally
called the vector and scalar potentials. However to date we are unable
measure either of these components directly, but can only measure their
differentials - eg the rate of change of scalar potential with distance
(= electric field) and the integral around a loop (ie "curl") of the
vector potential (= magnetic field). It turns out that when this charge
displacement field is propagating in a vacuum, these two components are
naturally perpendicular because they are orthogonal components (in a
mathematical sense) of the one entity. One might just as well ask "why
is length always perpendicular to breadth?". The answer would be simply
that it is a convenient way to measure and define two independent
components of a useful quantity called "area"!
To provide an intuitive illustration of an EM wave one might imagine a
long steel rod, one end of which is suddenly given a sharp torsional
jerk or twist. This torsional displacement wave is a pure shear wave as
there is no compression or rarefaction associated with it, and it will
propagate along the rod from one end to the other as a coherent entity
and at at characteristic speed determined only by the density of the
material and its shear modulus (spring coefficient). If the mass
displacement in the material is equated to the charge displacement in
the vacuum then (I think!) this becomes a very good analogy of an EM
wave propagating in a vacuum. The reason I have chosen torsional waves
is because as far as we know the vacuum only supports charge based shear
waves (ie displacement perpendicular to propagation. Experiments seem
to prove that the vacuum does not support charge based pressure waves -
ie displacement parallel to propagation as in sound waves - which is
very surprising and remarkable I think!)
If we now consider a small volume of the steel rod at its surface and
analyze the stresses and strains in that volume, then we can always
identify two conjugate quantities that between them support an
oscillation and due to their distributed nature support the wave
propagation. An analogous quantity to the vector potential (charge
proximity times its velocity per unit volume) I think would be the
linear momentum density (mass times velocity per unit volume). So the
analogous quantity to the magnetic field is the mathematical "curl" of
this - which is how much rotational component is present in this
momentum. This is very closely related to (and possibly exactly equal
to) the *angular* momentum density. The direction of angular momentum
is always specified by the axis about which the quantity is revolving -
and so in the case of this small volume at the surface of the rod, this
axis is perpendicular to the surface of the rod.
The analogous quantity to the electric field (or electric displacement)
I think would be the shear strain density (ie how much the material is
displaced in shear per unit distance along the rod and per unit
volume). This shear displacement of course occurs in a direction which
is tangential to the surface of the rod and about its axis - that being
the direction that we applied the initial jerk.
So here we have the magnetic field (angular momentum density) which is
perpendicular to the surface of the rod, and the electric field (shear
strain density) which is tangential to the surface of the rod, and both
of these two are perpendicular to the direction of propagation of the
wave along the axis of the rod. Now you can see that these two fields
are simply mathematically orthogonal energy components of the single
entity which is the wave motion. They are perpendicular only because of
the interacting components and their definitions that we have chosen to
describe the wave in terms of - in this case "angular momentum" (kinetic
energy) and "shear strain" (potential energy) components.
If we chose instead to describe an EM wave in terms of its vector
potential (*linear* momentum density) and its electric field (shear
strain density) then these components would still be mathematically
orthogonal but they would be *parallel* in space. (They must always
however be perpendicular to the direction of propagation because EM
radiation supports axes of polarization.) So the conjugate fields of an
EM wave are only perpendicular because we chose to define the kinetic
element in rotational units. If instead we chose linear units - such as
the more fundamental but less measurable "vector potential" or even the
(possibly one-day measurable) "time rate of change of vector potential",
then the oscillating fields of an EM wave would be parallel.
