Chapter 12

Electromagnetic Waves in Matter

12.1 Waves in Linear Media

In a linear, isotropic medium with permittivity $\epsilon = \epsilon_r\epsilon_0$ and permeability $\mu = \mu_r\mu_0$, the wave equation becomes:

$$\nabla^2\mathbf{E} = \mu\epsilon\frac{\partial^2\mathbf{E}}{\partial t^2} \implies v = \frac{1}{\sqrt{\mu\epsilon}} = \frac{c}{n}$$

where $n = \sqrt{\mu_r\epsilon_r}$ is the index of refraction. For most optical materials $\mu_r \approx 1$, so $n \approx \sqrt{\epsilon_r}$.

The amplitude ratio becomes $B_0 = nE_0/c$, and the intensity is$I = \frac{c\epsilon}{2}E_0^2 = \frac{1}{2}\epsilon v E_0^2$.

12.2 Dispersion

Real materials are dispersive: $n$ depends on frequency $\omega$. The phase velocity $v_p = \omega/k$ and group velocity $v_g = d\omega/dk$ differ:

$$v_p = \frac{c}{n(\omega)}, \qquad v_g = \frac{c}{n + \omega\,dn/d\omega}$$

The Sellmeier equation models dispersion in glass:

$$n^2(\lambda) = 1 + \sum_j \frac{B_j\lambda^2}{\lambda^2 - C_j}$$

12.3 Absorption & Complex Refractive Index

In a conducting medium, the wave equation gives a complex wave vector$\tilde{k} = k + i\kappa$, leading to an evanescent wave:

$$\tilde{\mathbf{E}} = E_0 e^{-\kappa z}e^{i(kz-\omega t)}\hat{x}$$

The field decays with the skin depth$\delta = 1/\kappa$:

$$\delta = \sqrt{\frac{2}{\mu\sigma\omega}} \qquad \text{(for good conductors: } \sigma \gg \epsilon\omega\text{)}$$

For copper at 60 Hz: $\delta \approx 8.5$ mm. At 1 GHz: $\delta \approx 2$ μm. This is why RF shielding works and why microwaves don't penetrate metal.

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