Difficulty: Intermediate | Prerequisites: Displacement Current and Maxwell's Equations notes, basic wave mechanics (amplitude, frequency, wavelength)
Once Maxwell's Equations predict that electromagnetic waves exist and travel at c, the next question is: what are these waves actually like? This topic covers the structure of an EM wave (perpendicular, in-phase E and B fields), how to read wave direction from the math, the energy and intensity those waves carry, and the Poynting vector that tracks energy flow. It also introduces the Doppler shift for light and the particle-wave duality of photons. This is where the abstract equations become the physics of sunlight, radio signals, and laser beams.
EM waves have perpendicular E and B fields oscillating in phase, propagating at c. They carry energy described by intensity I = ½cε₀E₀² and energy flow described by the Poynting vector S = (E × B)/μ₀. Light also exhibits Doppler shifts and has a dual wave-particle nature, with photon energy E = hf and momentum p = h/λ.
Electromagnetic (EM) wave
A self-propagating transverse wave consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of travel. In simple terms, it is a ripple in the electric and magnetic fields that carries energy through space at the speed of light.
Amplitude (E₀, B₀)
The peak value of the electric or magnetic field in the wave. The two amplitudes are linked by E₀ = cB₀.
Wave number (k)
The spatial frequency of the wave: k = 2π/λ. It tells you how many radians of oscillation fit per metre of space.
Angular frequency (ω)
The temporal frequency in radians per second: ω = 2π/T = 2πf. It tells you how fast the fields oscillate at a fixed point.
Poynting vector (S)
The vector S = (E × B)/μ₀ that points in the direction of energy flow and has magnitude equal to the instantaneous power per unit area carried by the wave. Think of it as the "energy current" of an electromagnetic wave.
Intensity (I)
The time-averaged power delivered per unit area: I = ⟨S⟩ = ½cε₀E₀². Units are W/m² (or equivalently kW/m²). This is what you measure with a light metre or a solar panel.
Energy density (u)
The energy stored per unit volume in the electromagnetic field. The instantaneous value is u = ε₀E², and the time-averaged value is ⟨u⟩ = ½ε₀E₀².
Doppler shift (relativistic)
The change in observed frequency of light due to relative motion between source and observer: f' = f√((1 + β)/(1 − β)), where β = v/c. Positive β means approach; negative means separation.
Redshift
The Doppler shift applied to astronomical objects moving away from the observer. The observed frequency is lower (wavelength longer, shifted towards red). Used to measure the recession speed of stars and galaxies.
Photon
A quantum of electromagnetic radiation. It carries energy E = hf and momentum p = h/λ, embodying the wave-particle duality of light. h = 6.63 × 10⁻³⁴ J·s (Planck's constant).
E and B fields are perpendicular to each other and perpendicular to the direction of propagation (transverse wave).
E and B oscillate in phase: they reach their peaks and zeros at the same points in space and time.
The direction of propagation is given by E × B.
Speed of light: c = ω/k = 1/√(μ₀ε₀) = E₀/B₀
E_x = E₀ cos(kz − ωt) and B_y = (E₀/c) cos(kz − ωt) for a wave travelling in the +z direction.
The argument (kz − ωt) indicates +z propagation; (kz + ωt) would indicate −z propagation.
The velocity relation: v = λf = ω/k
Amplitude: A (or E₀)
Wave number: k = 2π/λ
Wavelength: λ
Angular frequency: ω = 2π/T
Period: T
Frequency: f = 1/T
Velocity: v = λf = ω/k
E_x = E₀ sin(kz − ωt) propagates in the +z direction.
E_x = E₀ sin(ωt − kz) also propagates in the +z direction (same function, just a sign flip of the whole argument).
E_x = E₀ sin(ωz + kt) propagates in the −z direction.
The rule: if the spatial and temporal terms have opposite signs inside the argument, the wave moves in the +z direction. Same signs means −z.
Instantaneous energy density: u = ε₀E² (time-dependent, fluctuates with the wave)
Time-averaged energy density: ⟨u⟩ = ½ε₀E₀² (constant, useful for steady-state calculations)
The electric and magnetic contributions to the energy density are equal on average.
Intensity is the average energy delivered per unit time per unit area: I = (1/A)⟨dU/dt⟩ = Power/Area
For a plane wave: I = c⟨u⟩ = ½cε₀E₀²
For a point source radiating uniformly: the intensity at distance r falls as I = P/(4πr²), since the power spreads over the surface of a sphere.
Units: W/m² or kW/m²
S = (E × B)/μ₀
Direction: same as wave propagation (the direction energy flows).
Magnitude: |S| gives the instantaneous power per unit area.
Time-averaged magnitude: ⟨S⟩ = I = ½E₀²/(μ₀c)
Useful rearrangement: E₀ = √(2μ₀cI), which lets you find the field amplitude from a known intensity.
Relativistic Doppler formula: f' = f√((1 + β)/(1 − β))
β = v/c, where v is the relative speed between source and observer.
β > 0 (approaching): observed frequency increases (blueshift).
β < 0 (separating): observed frequency decreases (redshift).
Because of time dilation in special relativity, only the relative velocity matters. There is no distinction between "source moves" and "observer moves," unlike the acoustic Doppler effect.
In everyday situations v ≪ c, so β ≪ 1.
The formula simplifies to: f' ≈ f(1 + β)
This is adequate for most terrestrial applications.
Redshift is the Doppler shift applied to starlight from receding galaxies.
The observed wavelength is longer (redder) than the emitted wavelength.
Measuring redshift lets astronomers determine how fast objects are moving away, which is foundational to our understanding of the expanding universe.
Light behaves as both a wave and a particle.
As a particle: Energy E = hf, where h = 6.63 × 10⁻³⁴ J·s (Planck's constant).
As a wave: Momentum p = h/λ.
Higher frequency means higher energy per photon. This is why ultraviolet light causes sunburn but radio waves do not.
Quantity | Formula |
|---|---|
Wave speed | v = λf = ω/k = c |
E and B amplitude relation | E₀ = cB₀ |
Wave number | k = 2π/λ |
Angular frequency | ω = 2π/T = 2πf |
Instantaneous energy density | u = ε₀E² |
Average energy density | ⟨u⟩ = ½ε₀E₀² |
Intensity (plane wave) | I = ½cε₀E₀² |
Intensity (point source) | I = P/(4πr²) |
Poynting vector | S = (E × B)/μ₀ |
Field from intensity | E₀ = √(2μ₀cI) |
Relativistic Doppler | f' = f√((1 + β)/(1 − β)) |
Low-speed Doppler approx. | f' ≈ f(1 + β) |
Photon energy | E = hf |
Photon momentum | p = h/λ |
The Poynting vector and intensity are how engineers calculate the power output of antennas, the energy collected by solar panels, and the safe exposure limits for microwave and laser equipment. The Doppler shift of light is the basis for radar speed guns (using microwaves), medical Doppler ultrasound (same principle, different wave), and the measurement of how fast distant galaxies recede, which is how Hubble established that the universe is expanding.
"E and B are out of phase in an EM wave." They are in phase. Both reach their maximum and minimum values at the same point in space and time. Confusing this with the 90° spatial orientation (they are perpendicular in direction, not in phase).
"Intensity falls off as 1/r for a point source." Intensity falls as 1/r². The amplitude (field strength) falls as 1/r. Squaring the amplitude to get intensity introduces the extra factor of r.
"The Doppler shift for light works the same as for sound." For sound, it matters whether the source or the observer moves. For light (because of special relativity), only the relative velocity between source and observer matters.
"A photon has no momentum because it has no mass." A photon has momentum p = h/λ despite having zero rest mass. This is a direct consequence of special relativity: E² = (pc)² + (mc²)², and with m = 0, E = pc.
⚠️ Be able to write the standard form of E and B for a wave travelling in a given direction and identify which components are non-zero.
⚠️ Know how to compute intensity from E₀ (or B₀) and vice versa. The formula I = ½cε₀E₀² appears frequently.
⚠️ Expect a problem asking you to find the Poynting vector direction using the cross product E × B.
⚠️ The Doppler formula with β = v/c is testable. Know the sign convention: positive β for approach, negative for recession.
⚠️ Be ready to compute photon energy or momentum from a given wavelength or frequency.
⚠️ For point sources, remember that the relevant area is the surface of a sphere: A = 4πr².
True or False: E and B fields in an EM wave are perpendicular to each other but oscillate out of phase. (False; they are in phase.)
Fill in the blank: The direction of propagation of an EM wave is given by ____. (E × B)
True or False: For a point source, intensity is inversely proportional to r². (True)
Fill in the blank: Photon energy is given by E = ____. (hf)
True or False: In the relativistic Doppler effect, it matters whether the source or the observer is moving. (False; only relative velocity matters.)
Q: An EM wave has an electric field amplitude of E₀ = 500 V/m. What is the magnetic field amplitude and the intensity of the wave?
A: B₀ = E₀/c = 500/(3 × 10⁸) ≈ 1.67 × 10⁻⁶ T. Intensity I = ½cε₀E₀² = ½(3 × 10⁸)(8.85 × 10⁻¹²)(500²) ≈ 332 W/m².
Q: A star emits light at frequency f. An observer measures a lower frequency f' < f. Is the star moving towards or away from the observer?
A: The star is moving away. A lower observed frequency means redshift, which corresponds to β < 0 (source and observer separating).
Q: What is the direction of the Poynting vector for an EM wave with E in the x-direction and B in the y-direction?
A: S = (E × B)/μ₀. The cross product x̂ × ŷ = ẑ, so the Poynting vector (and wave propagation) is in the +z direction.
Q: A 100 W light bulb radiates uniformly in all directions. What is the intensity at a distance of 2 m?
A: I = P/(4πr²) = 100/(4π(2)²) = 100/(16π) ≈ 1.99 W/m².
Q: Calculate the energy of a photon with wavelength λ = 550 nm (green light).
A: f = c/λ = (3 × 10⁸)/(550 × 10⁻⁹) ≈ 5.45 × 10¹⁴ Hz. E = hf = (6.63 × 10⁻³⁴)(5.45 × 10¹⁴) ≈ 3.61 × 10⁻¹⁹ J ≈ 2.26 eV.
Energy density and intensity connect back to the capacitor energy (½ε₀E²) from earlier in the course; the same expression reappears as the instantaneous energy density in an EM wave. The Doppler shift for light generalises the acoustic Doppler effect covered in introductory mechanics. Photon energy E = hf is the starting point for quantum mechanics and photoelectric effect problems, which typically appear in PHY 214 or a modern physics course.
EM wave properties, electromagnetic wave, Poynting vector, intensity, energy density, wave propagation direction, E cross B, amplitude, wave number, angular frequency, wavelength, frequency, period, Doppler shift, relativistic Doppler, redshift, blueshift, photon energy, photon momentum, wave-particle duality, Planck's constant, point source intensity, inverse square law, PHY 212, UIUC physics, midterm 4 review