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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

Light travels through glass ( $n=1.50$ ). Find its speed (in m/s).

Your answer

m/s

2 of 7

Find the reflectance $R$ at normal incidence for air ( $n_1=1.0$ ) to glass ( $n_2=1.5$ ). $R=((n_1-n_2)/(n_1+n_2))^2$ .

Your answer

(dimensionless)

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