Electromagnetic Waves in Matter

When EM waves enter matter, they interact with the atoms and molecules. This changes their propagation speed and leads to phenomena like reflection and transmission at boundaries.

1. Propagation in Linear Media

In a linear, homogeneous dielectric material, the wave equation still holds, but the constants $\epsilon_0$ and $\mu_0$ are replaced by the permittivity $\epsilon$ and permeability $\mu$ of the material.

v = \frac{1}{\sqrt{\epsilon\mu}} = \frac{c}{n}

Where $n = \sqrt{\frac{\epsilon\mu}{\epsilon_0\mu_0}}$ is the Index of Refraction. Since most materials are nearly non-magnetic ($\mu \approx \mu_0$), we often use $n \approx \sqrt{\epsilon_r}$.

Interactive: Propagation in Media

2. Boundary Conditions (Normal Incidence)

When a wave hits an interface between two different media, some of it is reflected and some is transmitted. For normal incidence (hitting the surface straight on), the boundary conditions require:

Continuity The tangential components of $\vec{E}$ and $\vec{H}$ must be continuous across the boundary.

This leads to the Fresnel Equations for the reflection ($R$) and transmission ($T$) coefficients (for normal incidence):

E_{0R} = \frac{n_1 - n_2}{n_1 + n_2} E_{0I}, \quad E_{0T} = \frac{2n_1}{n_1 + n_2} E_{0I}

Note: If $n_2 > n_1$ (e.g., light entering glass), the reflected wave undergoes a $180^\circ$ phase shift.

Deep Dive: Boundary Conditions & Reflection