When light interacts with a dielectric medium, we have seen
that it induces an electric polarization, i.e. a density of electric dipole
moments. In turn, these dipoles create a field that affects the propagation of
light in the medium. This is reflected in Maxwell's equations for a dielectric
medium and consequently in the propagation equation for the electric
field (also called the "wave equation").
There are no free charges and electrical currents in a
dielectric medium. However, the electric field induces a polarization. So, in CGS
units, Maxwell's equations become:
(Eq.9.1a)
(Eq.9.1b)
(Eq.9.1c)
(Eq.9.1d)
where we used the same notations as in part 8 of this
tutorial. Also in CGS units, the polarization is defined as:
(Eq.9.2)
where
is the dielectric permittivity tensor.
Hence, from Eq. 9.1c we see that:
(Eq.9.3)
The fact that the divergence of E is zero will be very
useful
a few lines below.
We now look at the curl of the Eq. 9.1a:
(Eq.9.4)
Replacing with Eq. 9.2 for the value of the polarization we
can write:
(Eq.9.5)
On the other hand, we can also use the following formula,
which is general for any vector F:
(Eq.9.6)
Bearing in mind the result from Eq.9.3, namely that the divergence of E
is zero, this leads to:
(Eq.9.7)
In a final step, we can now combine Eqs. 9.5 and 9.7,
obtaining the propagation equation for the electric field:
(Eq.9.8)
Note that a very similar equation can be obtained for the
propagation of light in vacuum! The only difference for light propagating in a
dielectric medium is the present of the dielectric permittivity tensor.
