15.2 The Behavior of Electromagnetic Radiation

Properties and Behavior of Electromagnetic Radiation

Electromagnetic radiation changes speed and direction as it moves through different materials. The rules governing these changes, from Snell's law to the wave equation, let you predict exactly how light will behave at boundaries, in thin films, and across the full electromagnetic spectrum.

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Behavior of electromagnetic radiation

Electromagnetic radiation travels at different speeds depending on the medium. In a vacuum, it moves at the speed of light: $c \approx 3 \times 10^8 \text{ m/s}$. In materials like glass or water, it slows down. You can find the speed in any medium using:

$v = \frac{c}{n}$

where $n$ is the refractive index of the medium. A higher refractive index means a slower speed. For example, glass with $n = 1.5$ gives $v = \frac{3 \times 10^8}{1.5} = 2 \times 10^8 \text{ m/s}$.

Refraction occurs when electromagnetic radiation crosses from one medium into another and changes direction. The relationship between the incoming and outgoing angles is described by Snell's law:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

• $n_1$ and $n_2$ are the refractive indices of the first and second media
• $\theta_1$ is the angle of incidence (measured from the normal), and $\theta_2$ is the angle of refraction
• Light bends toward the normal when entering a denser medium (higher $n$) and away from the normal when entering a less dense medium

Total internal reflection happens when light travels from a higher-$n$ medium into a lower-$n$ medium and the angle of incidence exceeds the critical angle. Beyond this angle, no light passes through; it all reflects back. The critical angle is:

$\theta_c = \arcsin\left(\frac{n_2}{n_1}\right), \quad n_1 > n_2$

This is the principle behind fiber optics: light bounces along the inside of the fiber without escaping.

Diffraction occurs when electromagnetic waves encounter obstacles or pass through openings. The waves bend around edges and spread out, which becomes most noticeable when the obstacle or opening is comparable in size to the wavelength.

Wave equation calculations

The wave equation connects three fundamental properties of any electromagnetic wave:

$v = f\lambda$

• Frequency ($f$): the number of wave cycles per second, measured in hertz (Hz)
• Wavelength ($\lambda$): the distance between consecutive crests (or troughs), measured in meters
• Speed ($v$): how fast the wave propagates through the medium, in m/s

Given any two of these quantities, you can solve for the third. Here's the process:

1. Identify which two quantities you know.
2. Rearrange the equation: $f = \frac{v}{\lambda}$ or $\lambda = \frac{v}{f}$.
3. Plug in values and solve, keeping units consistent.

In a vacuum, the speed is always $c$, so the equation becomes $c = f\lambda$. For example, a radio station broadcasting at $f = 100 \text{ MHz} = 1 \times 10^8 \text{ Hz}$ has a wavelength of $\lambda = \frac{3 \times 10^8}{1 \times 10^8} = 3 \text{ m}$.

One thing to watch: frequency does not change when a wave enters a new medium. Only wavelength and speed change. So if you know the frequency in a vacuum, it's the same frequency inside glass or water.

The amplitude of an electromagnetic wave represents the maximum displacement of the electric (or magnetic) field from its equilibrium value. Amplitude relates to the wave's energy and intensity, not its speed or frequency.

Light polarization and interference

Polarization describes the orientation of the electric field oscillations in an electromagnetic wave. Unpolarized light (like sunlight) has electric fields vibrating in random directions perpendicular to the direction of travel. Light can also be:

• Linearly polarized: the electric field oscillates in a single plane
• Circularly polarized: the electric field direction rotates as the wave propagates

A polarizing filter transmits only the component of the electric field parallel to the filter's transmission axis, blocking everything else. When polarized light passes through a second filter (called an analyzer), the transmitted intensity follows Malus's law:

$I = I_0 \cos^2 \theta$

where $I_0$ is the intensity of the incoming polarized light and $\theta$ is the angle between the light's polarization direction and the filter's axis. At $\theta = 0°$, all light passes through. At $\theta = 90°$, none does.

Thin-film interference occurs when light reflects off both the top and bottom surfaces of a thin film (like a soap bubble or oil slick). These two reflected waves overlap and interfere:

• Constructive interference (bright fringe): the path difference equals a whole number of wavelengths, so the waves reinforce each other.
  • Condition: $2nd = m\lambda$, where $n$ is the film's refractive index, $d$ is the film thickness, $m$ is a positive integer, and $\lambda$ is the wavelength in vacuum.
• Destructive interference (dark fringe): the path difference equals a half-integer number of wavelengths, so the waves cancel.
  • Condition: $2nd = \left(m + \frac{1}{2}\right)\lambda$

Electromagnetic spectrum and dispersion

The electromagnetic spectrum covers all types of electromagnetic radiation, organized by wavelength (or equivalently, frequency). From longest wavelength to shortest:

• Radio waves ($\lambda > 1 \text{ m}$): communication, broadcasting
• Microwaves ($\lambda \sim 1 \text{ mm}$ to $1 \text{ m}$): radar, cooking
• Infrared ($\lambda \sim 700 \text{ nm}$ to $1 \text{ mm}$): thermal radiation, remote controls
• Visible light ($\lambda \sim 400$ to $700 \text{ nm}$): the narrow band human eyes detect, from violet to red
• Ultraviolet ($\lambda \sim 10$ to $400 \text{ nm}$): causes sunburn, used in sterilization
• X-rays ($\lambda \sim 0.01$ to $10 \text{ nm}$): medical imaging
• Gamma rays ($\lambda < 0.01 \text{ nm}$): nuclear processes, highest energy

All of these travel at $c$ in a vacuum. The only differences are wavelength and frequency, which are inversely related through $c = f\lambda$.

Dispersion occurs because a material's refractive index depends slightly on wavelength. Shorter wavelengths (violet) typically have a higher refractive index than longer wavelengths (red), so they bend more when passing through a prism. This separates white light into its component colors, producing a spectrum. Dispersion also explains why rainbows form: water droplets act like tiny prisms, refracting and reflecting sunlight at slightly different angles for each color.