momentum and energy are related by the equationU=p/c, where
pis the magnitude of the momentum vector, andU=Ue+Umis
the sum of the energy of the electric and magnetic fields. We can
now demonstrate this without explicitly referring to relativity, and
connect it to the specific structure of a light wave.
The energy density of a light wave is related to the magnitudes
of the fields in a specific way — it depends on the squares of their
magnitudes,E^2 andB^2 , which are the same as the dot products
E·EandB·B. We argued on page 604 that since energy is a
scalar, the only possible expressions for the energy densities of the
fields are dot products like these, multiplied by some constants. This
is because the dot product is the only mathematically sensible way
of multiplying two vectors to get a scalar result. (Any other way
violates the symmetry of space itself.)
How does this relate to momentum? Well, we know that if we
double the strengths of the fields in a light beam, it will have four
times the energy, because the energy depends on the square of the
fields. But we then know that this quadruple-energy light beam
must have quadruple the momentum as well. If there wasn’t this
kind of consistency between the momentum and the energy, then we
could violate conservation of momentum by combining light beams
or splitting them up. We therefore know that the momentum den-
sity of a light beam must depend on a field multiplied by a field.
Momentum, however, is a vector, and there is only one physically
meaningful way of multiplying two vectors to get a vector result,
which is the cross product (see page 1024). The momentum density
can therefore only depend on the cross productsE×E,B×B, and
E×B. But the first two of these are zero, since the cross product
vanishes when there is a zero angle between the vectors. Thus the
momentum per unit volume must equalE×Bmultiplied by some
constant,
dp= (constant)E×Bdv
This predicts something specific about the direction of propagation
of a light wave: it must be along the line perpendicular to the electric
and magnetic fields. We’ve already seen that this is correct, and also
that the electric and magnetic fields are perpendicular to each other.
Therefore this cross product has a magnitude
|E×B|=|E||B|sin 90◦
=|E||B|=|E|^2
c=c|B|^2 ,where in the last step the relation|E|=c|B|has been used.
We now only need to find one physical example in order to fix the
constant of proportionality. Indeed, if we didn’t know relativity, it
would be possible to believe that the constant of proportionality was
Section 11.6 Maxwell’s equations 731