Difficulty: Intermediate | Prerequisites: Maxwell's equations (conceptual), relationship between E and B fields, basic wave properties (wavelength, frequency, speed).
Tags: electromagnetic wave, EM wave, Poynting vector, intensity, Doppler shift, polarisation, Malus's law, speed of light, PHYS
Electromagnetic waves are the payoff of everything you have studied about electric and magnetic fields. Maxwell's equations predict that oscillating E and B fields sustain each other and propagate through space at the speed of light, carrying energy and momentum with them. This section covers how to describe that energy flow (the Poynting vector and intensity), how relative motion shifts the observed frequency (the relativistic Doppler effect), and how restricting the oscillation direction of the E field produces polarised light. These topics bridge the gap between circuits/fields and optics.
EM waves carry energy described by the Poynting vector S = (E × B)/μ₀, with intensity equal to power per unit area. The E and B amplitudes are linked by B₀ = E₀/c. Relative motion between source and observer shifts the frequency via the relativistic Doppler formula. Polarisers reduce the intensity of light: unpolarised light loses half its intensity through the first polariser, and Malus's law (I = I₀ cos²θ) governs every subsequent one.
Poynting vector (S)
S = (E × B)/μ₀. It points in the direction the electromagnetic wave is travelling and its magnitude gives the instantaneous power per unit area (watts per square metre) carried by the wave. Think of it as an arrow that says "energy is flowing this way, this fast."
Intensity (S or I)
The time-averaged power per unit area delivered by the wave: I = Power / Area. For a sinusoidal EM wave, the average intensity relates to the peak fields by I = E₀B₀/(2μ₀) = E₀²/(2μ₀c). In simple terms, intensity is how bright or strong the wave is at a given point.
Speed of light relationship
The peak electric and magnetic field amplitudes in an EM wave are not independent. They are locked together by B₀ = E₀/c, where c ≈ 3 × 10⁸ m/s. If you know one, you know the other.
Angular frequency and wave number
ω = 2πf (angular frequency) and the wave speed relates to wavelength by c = λf. The wave number k = 2π/λ, and c = ω/k. These are the standard wave relationships, applied to light.
Doppler shift (relativistic, for EM waves)
The change in observed frequency when the source and observer move relative to one another. Unlike the acoustic Doppler effect, the EM version uses the relativistic formula because light always travels at c regardless of the motion of source or observer.
Linear polarisation
A condition in which the electric field vector of an EM wave oscillates in a single plane. Unpolarised light has E vectors pointing in all random directions perpendicular to propagation. A polariser filters it down to one direction.
Malus's law
I_f = I₀ cos²θ, where θ is the angle between the incoming polarisation direction and the polariser's transmission axis. This applies when already-polarised light hits a polariser. In simple terms, the closer the light's polarisation aligns with the filter, the more light gets through.
An EM wave consists of oscillating E and B fields, perpendicular to each other and perpendicular to the direction of propagation.
The direction of propagation is given by the Poynting vector:
S = (E × B) / μ₀
The magnitude of S gives the instantaneous intensity (power per unit area).
The time-averaged intensity for a sinusoidal wave is:
I = Power / Area
Also expressible as I = E₀²/(2μ₀c) or I = c ε₀ E₀²/2
The amplitudes are related by:
B₀ = E₀ / c
At any instant, E and B are in phase (both peak at the same time, both zero at the same time) and perpendicular to each other.
If you know the direction of E and the direction of propagation, you can find the direction of B using the right-hand rule (since S = E × B / μ₀ must point in the propagation direction).
ω = 2πf (angular frequency in rad/s)
c = λf (wave speed equals wavelength times frequency)
c = ω / k (wave speed from angular frequency and wave number)
These are identical to the relationships for any wave; the special feature of EM waves in vacuum is that c is always 3 × 10⁸ m/s.
For a source and observer moving apart (increasing separation):
f' = f √((1 − β) / (1 + β))
where β = v/c (the relative speed as a fraction of the speed of light)
This is a redshift: the observed frequency is lower than the emitted frequency.
For a source and observer moving toward each other, swap the signs under the square root (or equivalently, use β as negative), giving a blueshift (higher observed frequency).
At everyday speeds (v ≪ c), β is tiny and the shift is very small. The relativistic formula matters for astronomical objects and high-speed particles.
Unpolarised light through a polariser:
The first polariser always cuts the intensity in half:
I_f = ½ I₀
The transmitted light is now linearly polarised along the polariser's axis.
Polarised light through a second polariser (Malus's law):
I_f = I₀ cos²θ
θ is the angle between the polarisation direction of the incoming light and the transmission axis of the polariser.
At θ = 0°: all light passes (I_f = I₀).
At θ = 90°: no light passes (I_f = 0). The polarisers are "crossed."
For chains of multiple polarisers, apply each step sequentially:
First polariser: I₁ = ½ I₀ (if starting from unpolarised)
Each subsequent polariser: I_next = I_prev cos²θ, where θ is the angle between the previous polarisation direction and the next polariser's axis.
Quantity | Expression |
|---|---|
Poynting vector | S = (E × B) / μ₀ |
Intensity | I = Power / Area |
E-B amplitude relation | B₀ = E₀ / c |
Angular frequency | ω = 2πf |
Wave speed | c = λf = ω/k |
Doppler shift (increasing separation) | f' = f √((1 − β)/(1 + β)), β = v/c |
Unpolarised → polariser | I_f = ½ I₀ |
Malus's law (polarised light) | I_f = I₀ cos²θ |
The Poynting vector is how engineers calculate the power delivered by an antenna or the energy flux from a laser. The Doppler shift for EM waves is the basis of how astronomers measure the speed at which stars and galaxies are moving toward or away from us (redshift/blueshift). Polarisation is used in sunglasses (which block horizontally polarised glare from flat surfaces), LCD screens (which use liquid crystals to rotate polarisation between two crossed polarisers), and 3D cinema glasses (which deliver different polarisations to each eye).
"The Poynting vector points along the E field." It does not. S = E × B / μ₀ is perpendicular to both E and B. It points in the direction the wave travels.
"Unpolarised light through a polariser follows Malus's law." Malus's law (cos²θ) applies only to already-polarised light. For unpolarised light hitting the first polariser, the rule is simply I_f = ½ I₀, regardless of the polariser's orientation.
"The EM Doppler formula is the same as the sound Doppler formula." It is not. Sound has separate cases for moving source and moving observer because sound travels through a medium. EM waves in vacuum have no medium, so only relative velocity matters, and the formula is relativistic.
"E and B are out of phase in an EM wave." In free space they are in phase: both reach their maximum and both pass through zero at the same times.
⚠️ Poynting vector direction problems often appear as "which way is the wave travelling?" given E and B directions. Use the cross product or right-hand rule.
⚠️ Polarisation chain problems (2 or 3 polarisers in sequence) are a favourite. Remember: first polariser halves the intensity, then apply cos²θ at each subsequent step. Track the new polarisation direction after each polariser.
⚠️ The E-B relationship B₀ = E₀/c is a quick calculation that often appears as a "gimme" part of a larger problem. Do not forget it.
⚠️ For Doppler problems, clearly identify whether the source and observer are moving apart or together before choosing the sign in the formula.
Fill in the blank: The Poynting vector S = (E × B) / ___.
True or false: B₀ = E₀ × c.
Fill in the blank: Unpolarised light passing through a single polariser has its intensity reduced to ___ of the original.
True or false: At θ = 90°, Malus's law gives maximum transmitted intensity.
Fill in the blank: The EM Doppler shift for increasing separation gives a ___ (higher/lower) observed frequency.
Answers: 1. μ₀. 2. False (B₀ = E₀/c). 3. One-half (½). 4. False (it gives zero). 5. Lower (redshift).
Q: An EM wave has an electric field amplitude of E₀ = 600 V/m. What is the magnetic field amplitude?
A: B₀ = E₀/c = 600 / (3 × 10⁸) = 2 × 10⁻⁶ T = 2 µT.
Q: Unpolarised light of intensity 80 W/m² passes through two polarisers. The second polariser is oriented at 30° to the first. What is the final intensity?
A: After the first polariser: I₁ = ½ × 80 = 40 W/m². After the second: I₂ = 40 × cos²(30°) = 40 × (√3/2)² = 40 × 0.75 = 30 W/m².
Q: A source emits light at frequency f. An observer moves away from the source at 0.1c. Is the observed frequency higher or lower, and by roughly how much?
A: Lower (redshift). f' = f √((1 − 0.1)/(1 + 0.1)) = f √(0.9/1.1) = f √(0.818) ≈ 0.905f. The observed frequency is about 90.5% of the original.
Q: The electric field in an EM wave points in the +y direction and the wave travels in the +x direction. What direction does the magnetic field point?
A: S is along +x and E is along +y. Since S ∝ E × B, we need ŷ × B̂ = x̂, which gives B̂ = +ẑ. The magnetic field points in the +z direction.
Q: Why does inserting a third polariser between two crossed polarisers allow some light through, when without it the transmission is zero?
A: The middle polariser, oriented at an angle between the two crossed axes, transmits a component of the polarised light from the first polariser (by cos²θ₁) and rotates the polarisation direction. The light arriving at the final polariser is no longer perpendicular to it, so a component gets through (by cos²θ₂). Without the middle polariser, the polarisation from the first is exactly perpendicular to the second, and cos²(90°) = 0.
The Poynting vector and EM wave energy connect back to the energy stored in inductors (½LI², magnetic field energy) and capacitors (½CV², electric field energy) you studied in circuit theory. Polarisation leads into optics topics such as Brewster's angle, thin-film interference, and birefringence. The Doppler shift for EM waves connects to special relativity and is a cornerstone of astrophysics (measuring stellar velocities, confirming the expansion of the universe). If your course continues into modern physics, the quantisation of EM wave energy (photons, E = hf) builds directly on the wave description here.
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