All electromagnetic waves travel at c = 3 × 10⁸ m/s in vacuum but differ in wavelength and frequency.
From longest to shortest wavelength: radio → microwave → infrared → visible → ultraviolet → X-ray →
gamma ray.
Radio
λ > 1 m, f < 300 MHz
Microwave
1 mm – 1 m, WiFi/5G/radar
Infrared
700 nm – 1 mm, heat/thermal
Visible
400–700 nm (violet→red)
Ultraviolet
10–400 nm, sunburn
X-ray
0.01–10 nm, medical imaging
Gamma
< 0.01 nm, nuclear decay
c = fλ
c = 3 × 10⁸ m/s, f = frequency (Hz)λ =
wavelength (m)Higher frequency = shorter wavelength = more energy
Worked Example 1
WiFi Signal Wavelength
Problem: Your WiFi router broadcasts at 5.0 GHz. What is
the wavelength? What part of the EM spectrum is this?
Apply c = fλ
λ = c/f = 3×10⁸ / 5.0×10⁹
λ = 0.06 m = 6 cm (microwave)
Answer: 6 cm wavelength - microwave range. This is why
WiFi can be blocked by walls (wavelength comparable to gaps) and why microwave ovens operate at
similar frequencies (2.45 GHz, λ = 12.2 cm).
2 Maxwell's Equations - The Complete Theory
James Clerk Maxwell unified electricity, magnetism, and optics into four elegant equations that
describe all electromagnetic phenomena. They predict that changing electric fields create
magnetic fields and vice versa - together forming self-propagating waves at the speed of light.
Gauss (E)
∇·E = ρ/ε₀ (charges create E fields)
Gauss (B)
∇·B = 0 (no magnetic monopoles)
Faraday
∇×E = −∂B/∂t (changing B creates E)
Ampère-Maxwell
∇×B = μ₀J + μ₀ε₀∂E/∂t
3 Wave Properties - Interference &
Diffraction
EM waves exhibit interference (constructive when crests align, destructive when crest meets
trough) and diffraction (bending around obstacles or through slits). These wave behaviors prove
light is a wave - complementing its particle nature (wave-particle duality).
Young's Double Slit: d sin θ = mλ
d = slit separation, θ = angle to bright fringem
= 0, ±1, ±2... (fringe order)Bright fringes where path difference = whole
wavelengths
Worked Example 2
Double Slit Experiment
Problem: Light of wavelength 600 nm passes through two
slits 0.2 mm apart. A screen is 2 m away. What is the spacing between bright fringes?
Fringe spacing formula
Δy = λL/d = (600×10⁻⁹ × 2) / (0.2×10⁻³)
Δy = 6.0 × 10⁻³ m = 6.0 mm
Answer: 6.0 mm between bright fringes. Longer wavelength
→ wider spacing. Narrower slits → wider spacing. This experiment was first performed by Thomas
Young in 1801, proving light is a wave.
4 Polarization
Light is a transverse wave - the electric field oscillates perpendicular to the direction of
travel. Unpolarized light has E oscillating in all perpendicular directions. A polarizer
transmits only one direction. Two perpendicular polarizers block all light.
Malus's Law: I = I₀ cos²θ
I₀ = intensity after first polarizerθ = angle
between polarizer axesAt 90°: I = 0 (crossed polarizers)
Worked Example 3
Malus's Law - Polarizer Angle
Problem: Unpolarized light passes through two
polarizers. The second is at 30° to the first. What fraction of the original intensity passes
through?
Two-stage reduction
After 1st polarizer: I₁ = I₀/2 (always halves unpolarized)
After 2nd: I₂ = I₁ cos²(30°) = (I₀/2)(√3/2)² = (I₀/2)(3/4)
I₂ = 3I₀/8 = 0.375 I₀ (37.5%)
Answer: 37.5% of original intensity. At 45° it would be
25%, at 60° only 12.5%, at 90° zero. Polarized sunglasses work by blocking horizontally
polarized glare from reflective surfaces.
5 Electromagnetic Wave Energy
EM waves carry energy described by the Poynting vector. The intensity (power per area) is
proportional to the square of the electric field amplitude.
I = P/A = ½cε₀E₀²
I = intensity (W/m²)E₀ = peak electric field
(V/m)Sun at Earth: I ≈ 1361 W/m² → E₀ ≈ 1013 V/m
6 Inverse Square Law & Radiation
EM radiation spreads over an expanding sphere. Intensity decreases with the square of distance
from the source.
I = P / (4πr²)
Double the distance → ¼ the intensityTriple the
distance → 1/9 the intensity
Why stars are dim:
The Sun's luminosity is 3.85 × 10²⁶ W. At Earth (1.5 × 10¹¹ m away): 1,361 W/m². At Pluto:
only 0.9 W/m². At the nearest star (4.24 ly): 3 × 10⁻⁸ W/m².