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Topic 8 of 9

Electromagnetic Waves in Matter

Electromagnetic waves propagate through matter as well as vacuum, but with a modified speed $v=c/n$ where $n=\sqrt{\varepsilon_r\mu_r}$ is the index of refraction. At boundaries between media, waves are partially reflected and partially transmitted according to the Fresnel equations. Dispersion — the frequency-dependence of $n$ — is the origin of rainbows.

Key Concepts

Wave in Medium

Speed

v=c/n

, wavelength

\lambda=\lambda_0/n

, frequency unchanged. Amplitude decays in lossy media as

e^{-\alpha z/2}

where

\alpha

is the absorption coefficient.

Index of Refraction

n=\sqrt{\varepsilon_r\mu_r}\approx\sqrt{\varepsilon_r}

(for non-magnetic media). Depends on frequency (dispersion). Snell's law:

n_1\sin\theta_1=n_2\sin\theta_2

Fresnel Equations

Reflection and transmission amplitudes at normal incidence:

r=(n_1-n_2)/(n_1+n_2)

t=2n_1/(n_1+n_2)

. Reflectance:

R=r^2=[(n_1-n_2)/(n_1+n_2)]^2

Total Internal Reflection

When

n_1>n_2

, light can't escape if

\theta_1>\theta_c=\arcsin(n_2/n_1)

. This is the basis of optical fibers.

Key Equations

Wave speed in medium

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

Speed of EM wave in medium with permittivity ε and permeability μ.

Fresnel (normal incidence)

R = \left(\frac{n_1-n_2}{n_1+n_2}\right)^2, \quad T = 1-R

Reflectance R and transmittance T at a planar interface.

Critical angle

\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)

Angle of total internal reflection; exists only when n₁ > n₂.

Worked Example

Reflection at Glass-Air Interface

Problem

Light travels from glass ( $n_1=1.5$ ) to air ( $n_2=1.0$ ) at normal incidence. Find the reflectance $R$ and the critical angle $\theta_c$ .

Solution

R = \left(\frac{1.5-1.0}{1.5+1.0}\right)^2 = \left(\frac{0.5}{2.5}\right)^2 = (0.20)^2 = 0.04 = 4\%

\theta_c = \arcsin(n_2/n_1) = \arcsin(1.0/1.5) = \arcsin(0.667) = 41.8°

Answer

R=4\%

;

\theta_c=41.8°

Practice

Exercises

7 problems

1 of 7

Drag the slider — the wave on the right slows down and its wavelength shrinks by factor $n$. Find the speed v = c/n in glass ($n = 1.50$) in m/s (enter 2e8).

wave in medium

Index of refraction n 1.80 → v = 1.667×10⁸ m/s

Drag n — the wave on the right slows AND its wavelength shrinks by factor n. Frequency stays the same.

Wave speed at n = 1.50

v = m/s

Enter 2e8 or 200000000 (m/s)

2 of 7

Drag $n_2$ and watch the reflected ray grow brighter. At air→glass ($n_1=1.0$, $n_2=1.5$), find the reflectance R = ((n₁−n₂)/(n₁+n₂))² (enter as decimal, e.g. 0.04).

fresnel reflection

Glass index n₂ 2.00 → R = 0.1111

Drag n₂ — as the contrast increases, more light reflects. At n₁=1.0 (air) to n₂=1.5 (glass), only 4% reflects.

Reflectance at n₂ = 1.5

R = (dimensionless)

R = ((1.0−1.5)/(1.0+1.5))² = ? Enter as decimal (e.g. 0.04)

3 of 7

Critical angle for total internal reflection from glass ( $n_1=1.5$ ) to air ( $n_2=1.0$ ). $\theta_c=\arcsin(n_2/n_1)$ (in degrees).

Unlock Exercise 3