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Dynamics and Reflection/Refraction

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 $\mbox{\boldmath$E$}$):

$\displaystyle \epsilon (\mbox{\boldmath$E$}_0 + \mbox{\boldmath$E$}_0'')\cdot\hat{\mbox{\boldmath$n$}}$ $\textstyle =$ $\displaystyle \epsilon'
\mbox{\boldmath$E$}_0'\cdot\hat{\mbox{\boldmath$n$}}$ (9.61)
$\displaystyle (\mbox{\boldmath$E$}_0 + \mbox{\boldmath$E$}_0'') \times \hat{\mbox{\boldmath$n$}}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$E$}_0' \times \hat{\mbox{\boldmath$n$}}$ (9.62)

where the latter cross product is just a fancy way of finding $E_\perp$ components. In most cases one wouldn't actually ``do'' this decomposition algebraically, one would just inspect the problem and write down the $\vert\vert$ and $\perp$ components directly using a sensible coordinate system (such as one where $\hat{\mbox{\boldmath$n$}} = \hat{\mbox{\boldmath$z$}}$).

Similarly for $\mbox{\boldmath$B$}$:

$\displaystyle (\mbox{\boldmath$B$}_0 + \mbox{\boldmath$B$}_0'') \cdot \hat{\mbox{\boldmath$n$}}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$B$}_0'\cdot\hat{\mbox{\boldmath$n$}}$ (9.63)
$\displaystyle \frac{1}{\mu}(\mbox{\boldmath$B$}_0 + \mbox{\boldmath$B$}_0'') \times \hat{\mbox{\boldmath$n$}}$ $\textstyle =$ $\displaystyle \frac{1}{\mu'}\mbox{\boldmath$B$}_0' \times \hat{\mbox{\boldmath$n$}}$ (9.64)

(where, recall, $\mbox{\boldmath$B$} = (\mbox{\boldmath$k$} \times \mbox{\boldmath$E$})/(vk)$ 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:
\begin{displaymath}
\vert\mbox{\boldmath$B$}_0\vert = \frac{\vert\mbox{\boldmath...
...vert}{v} = \sqrt{\mu \epsilon}\vert\mbox{\boldmath$E$}_0\vert
\end{displaymath} (9.65)



Subsections
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Next: Coordinate choice and Brewster's Up: Reflection and Refraction at Previous: Kinematics and Snell's Law   Contents
Robert G. Brown 2014-08-19