Introduction to Electrodynamics/Chapter 8/2/3
8.2.3 Propagation Through Linear Media
When we move out of vacuum and into linear media, all we have to do is switch a few constants and all the equations are the same.
Hi, this is Jonathan Gardner. This is a very short video on section 8.2.3 of Griffith's Introduction to Electrodynamics. We're covering how monochromatic plane waves behave in linear media as opposed to a vacuum. Previously, we calculated the energy, the Poynting vector, the momentum, and the intensity of a plane wave in a vacuum.
Velocity
Recall from earlier where we used Maxwell's Equations in linear media and found that it produces a wave function when there is no free charge or free currents. The velocity of the wave was simply:
<math>
v = c / n = 1/\sqrt{\epsilon \mu} </math>
where epsilon and mu are the permittivity and permeability of the linear media.
n is the index of refraction, which Snell's Law uses. We're going to show how waves reflect and transmit through boundaries shortly.
n is equal to <math>\sqrtTemplate:\epsilon \mu \over \epsilon 0 \mu 0</math>.
Let's write down the equations we had for energy, energy flux rate, momentum, and intensity, substituting epsilon for epsilon_0, mu for mu_0, and c/n for c.
<math> \begin{align} U &= \epsilon E_0^2 cos(\kappa x - \omega t + \delta) \\ \vec{S} &= {c \over n} U \hat{i} \\ \vec{p} &= {n \over c} U \hat{i} \\ I &= {1 \over 2} \epsilon {c \over n} E_0^2 \hat{i} \\ \end{align} </math>
Easy peasy.
Boundary Conditions
I'm going to preview the next sections with a re-iteration of the boundary conditions between linear media.
Rule 1: The E field in the perpendicular direction jumps:
<math>\epsilon_1 E_{1\perp} = \epsilon_2 E_{2\perp}\;</math>
The reason is simple: Take an infinitessimally small Gaussian box that includes the surface. The D field (which is just epsilon-E) "jumps" by the surface charge at the boundary. Here, we have no free charge.
Rule 2: The B field perpendicular to the boundary is constant.
<math>B_{1\perp} = B_{2\perp}\;</math>
If there was going to be a discontinuity, it would have to be due to a magnetic monopole.
Rule 3: The parallel E fields don't change.
<math>\vec{E_{1\parallel}} = \vec{E_{2\parallel}}\;</math>
The reason for this is likewise simple. If you take any path integral which passes through the boundary, then you can make it so small that there is no magnetic flux.
Finally, Rule 4: The parallel B fields jump.
<math>{1 \over \mu_1} \vec{B_{1\parallel}} = {1 \over \mu_2} \vec{B_{2\parallel}}\;</math>
If you take an infinitesimally small Amperian loop that covers the boundary, then the H fields (which are just B/mu) jumps by the amount of free surface current. Here, we have none.
These four rules are enough to describe completely what happens to an EM waves that passes through a boundary. These will help us relate the magnetic and electric fields on either side of the boundary.
In the next section, we're going to cover what happens when a wave hits head on. The section after that is what happens when a wave hits at an angle.
Let me know if you have questions in the comments below, or make a video response.
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