Now we do the dynamics, that is to say, the real physics. Real physics
is associated with the *equations of motion* of the EM field, that
is, with Maxwell's equations, which in turn become the wave equation, so
dynamics is associated with the boundary value problem satisfied by the
(wave equation) PDEs.

So what *are* those boundary conditions? Recall that the electric
*displacement* perpendicular to the surface must be continuous, that
the electric *field* parallel to the surface must be continuous,
that the magnetic *field* parallel to the surface must be continuous
and the magnetic *induction* perpendicular to the surface must be
continuous.

To put it another (more physical) way, the perpendicular components of
the electric field will be discontinous at the surface due to the
surface charge layer associated with the local polarization of the
medium in response to the wave. This polarization is actually *not
instantaneous*, and is a *bulk response* but here we will assume
that the medium can react instantly as the wave arrives and that the
wavelength includes many atoms so that the response is a collective one.
These assumptions are valid for e.g. visible light incident on ordinary
``transparent'' matter. Similarly, surface current loops cause magnetic
induction components parallel to the surface to be discontinuously
changed.

Algebraically, this becomes (for
):

(9.61) | |||

(9.62) |

where the latter cross product is just a fancy way of finding components. In most cases one wouldn't actually ``do'' this decomposition algebraically, one would just inspect the problem and write down the and components directly using a sensible coordinate system (such as one where ).

Similarly for
:

(9.63) | |||

(9.64) |

(where, recall, etc.) Again, one usually would not use this cross product algebraically, but would simply formulate the problem in a convenient coordinate system and take advantage of the fact that:

(9.65) |

- Coordinate choice and Brewster's Law
- Perpendicular to Plane of Incidence
- Parallel to Plane of Incidence
- Intensity
- Polarization Revisited: The Brewster Angle