Sections#

Review & Summary#

Electromagnetic Waves#

An electromagnetic wave consists of oscillating electric and magnetic fields. The various possible frequencies form a spectrum, a small part of which is visible light. A wave traveling along the \(x\) axis has \(E = E_m \sin(kx - \omega t)\) and \(B = B_m \sin(kx - \omega t)\), where \(E_m\) and \(B_m\) are the amplitudes. The oscillating fields induce each other. The speed in vacuum is \(c = 1/\sqrt{\mu_0\varepsilon_0}\), and the simultaneous magnitudes satisfy \(E = cB\).

Energy Flow#

The rate per unit area at which energy is transported is given by the Poynting vector \(\vec{S} = \vec{E} \times \vec{B}/\mu_0\). The direction of \(\vec{S}\) is perpendicular to both \(\vec{E}\) and \(\vec{B}\). The intensity \(I\) is the time-averaged magnitude:

(225)#\[ I = S_{\mathrm{avg}} = \frac{E_{\mathrm{rms}}^2}{c\mu_0} \]

where \(E_{\mathrm{rms}} = E_m/\sqrt{2}\). A point source emits waves isotropically. The intensity at distance \(r\) from a source of power \(P_s\) is

(226)#\[ I = \frac{P_s}{4\pi r^2} \]

Radiation Pressure#

When a surface intercepts electromagnetic radiation, a force is exerted. If the radiation is totally absorbed, \(F = IA/c\), where \(I\) is the intensity and \(A\) is the area perpendicular to the path. If totally reflected back along its original path, \(F = 2IA/c\). The radiation pressure is \(p_r = F/A\): thus \(p_r = I/c\) (absorbed) or \(2I/c\) (reflected).

Polarization#

Electromagnetic waves are polarized if their electric field vectors all lie in a single plane (the plane of oscillation). Light from common sources is unpolarized (randomly polarized). A polarizing sheet transmits only components parallel to its polarizing direction. For initially unpolarized light, \(I = \frac{1}{2}I_0\). For initially polarized light at angle \(\theta\) to the polarizing direction, Malus’s law: \(I = I_0 \cos^2\theta\).

Reflection and Refraction#

When a light ray encounters a boundary between two transparent media, a reflected ray and a refracted ray appear. The angle of reflection equals the angle of incidence. Snell’s law: \(n_1 \sin\theta_1 = n_2 \sin\theta_2\), where \(n_1\) and \(n_2\) are the indices of refraction.

Total Internal Reflection#

When \(n_1 > n_2\) and the angle of incidence exceeds the critical angle \(\theta_c = \arcsin(n_2/n_1)\), all light is reflected back into medium 1.

Polarization by Reflection#

At Brewster’s angle \(\theta_B = \arctan(n_2/n_1)\), the reflected light is fully polarized perpendicular to the plane of incidence.