Electromagnetic Waves

Wave Equation from Maxwell

In vacuum, taking curl of Faraday's law and using Ampere-Maxwell:

$\nabla^2\vec{E} = \mu_0\epsilon_0\frac{\partial^2\vec{E}}{\partial t^2}$

$\nabla^2\vec{B} = \mu_0\epsilon_0\frac{\partial^2\vec{B}}{\partial t^2}$

These are wave equations with speed:

$c = \frac{1}{\sqrt{\mu_0\epsilon_0}} = 3.00 \times 10^8 \text{ m/s}$

(Speed of light!)

Plane Wave Solution

Electric field: $\vec{E} = E_0\sin(kx - \omega t)\hat{j}$

Magnetic field: $\vec{B} = B_0\sin(kx - \omega t)\hat{k}$

where:

• $k = 2\pi/\lambda$ (wave number)
• $\omega = 2\pi f$ (angular frequency)
• $\omega = ck$ (dispersion relation)

Properties of EM Waves

1. Transverse: $\vec{E}$ and $\vec{B}$ perpendicular to propagation direction

2. Perpendicular to each other: $\vec{E} \perp \vec{B}$

3. In phase: $\vec{E}$ and $\vec{B}$ oscillate together

4. Right-hand rule: $\vec{E} \times \vec{B}$ points in propagation direction

5. Field magnitudes related: $\frac{E}{B} = c$

Energy in EM Wave

Energy density: $u = \epsilon_0 E^2 = \frac{B^2}{\mu_0}$

(Equal contributions from $\vec{E}$ and $\vec{B}$)

Intensity (average power per area): $I = \langle S \rangle = \frac{1}{2}\epsilon_0 cE_0^2 = \frac{E_0^2}{2\mu_0 c}$

or in terms of $B_0$: $I = \frac{cB_0^2}{2\mu_0}$

Momentum and Radiation Pressure

EM wave carries momentum:

$p = \frac{U}{c}$

Momentum density: $g = \frac{u}{c} = \frac{S}{c^2}$

Radiation pressure:

Complete absorption: $P = \frac{I}{c}$

Complete reflection: $P = \frac{2I}{c}$

Polarization

Linear polarization: $\vec{E}$ oscillates in fixed plane

Circular polarization: $\vec{E}$ rotates, constant magnitude

Unpolarized: Random polarization directions

Malus's law: Intensity through polarizer: $I = I_0\cos^2\theta$

where $\theta$ is angle from polarization axis.

Electromagnetic Spectrum

All EM waves travel at $c$ in vacuum, differ only in frequency/wavelength:

• Radio waves: $f < 10^9$ Hz
• Microwaves: $10^9$ - $10^{12}$ Hz
• Infrared: $10^{12}$ - $10^{14}$ Hz
• Visible: $4 \times 10^{14}$ - $7 \times 10^{14}$ Hz
• Ultraviolet: $10^{15}$ - $10^{17}$ Hz
• X-rays: $10^{17}$ - $10^{19}$ Hz
• Gamma rays: $f > 10^{19}$ Hz

Standing EM Waves

Boundary conditions (e.g., in cavity) create standing waves:

$E_y = 2E_0\sin(kx)\cos(\omega t)$

Nodes: $E = 0$ at $x = 0, \lambda/2, \lambda, ...$

Used in:

• Lasers
• Microwave ovens
• Radio antennas

Doppler Effect

Source moving with velocity $v$:

Moving toward observer: $f' = f\frac{c}{c - v}$

Moving away: $f' = f\frac{c}{c + v}$

For $v \ll c$: $\frac{\Delta f}{f} \approx \frac{v}{c}$