Describe the electromagnetic spectrum in terms of frequency, wavelength and photon energy, and outline how Maxwell's equations conceptually predict electromagnetic waves travelling at the speed of light

What this dot point is asking

NESA wants you to know the layout of the electromagnetic spectrum, the relationships $c = f \lambda$ and $E = hf$, and the historical and conceptual significance of Maxwell's equations. You should be able to identify each band of the spectrum, compare wavelengths, frequencies and photon energies across the bands, and explain why Maxwell's prediction unified optics and electromagnetism.

The answer

The spectrum

Electromagnetic (EM) radiation is a transverse wave of oscillating electric and magnetic fields propagating at the speed of light, $c = 2.998 \times 10^8$ m/s in vacuum. The fields are perpendicular to each other and to the direction of propagation. EM waves do not need a medium. The diagram shows the seven bands ordered by wavelength, with the visible spectrum sitting between ultraviolet and infrared.

The spectrum, ordered from longest wavelength to shortest:

Visible light runs from violet ($\sim 400$ nm) to red ($\sim 700$ nm). UV beyond about $10$ eV and X-rays ionise atoms; radio, microwave, infrared and most visible photons cannot.

Key relationships

For a wave of frequency $f$ and wavelength $\lambda$ travelling at speed $c$:

The photon energy (the smallest "packet" of EM energy at frequency $f$) is:

where $h = 6.626 \times 10^{-34}$ J s is Planck's constant. Higher-frequency, shorter-wavelength radiation carries more energy per photon.

Maxwell's equations, in words

By 1865, James Clerk Maxwell had combined four laws of electromagnetism into a self-consistent set:

1. Gauss's law for electricity. Electric field lines start on positive charges and end on negative charges; the total flux through a closed surface is proportional to the enclosed charge.
2. Gauss's law for magnetism. Magnetic field lines form closed loops; no magnetic monopoles exist.
3. Faraday's law of induction. A changing magnetic flux produces a circulating electric field (the EMF driving induced currents).
4. The Ampere-Maxwell law. A current and a changing electric flux both produce a circulating magnetic field. Maxwell's added term (the displacement current) was the key insight.

Together, items 3 and 4 say each kind of changing field creates the other. Combining them mathematically gives a wave equation for $\vec{E}$ and $\vec{B}$ that propagates at:

Substituting the static, table-book values $\mu_0 = 4\pi \times 10^{-7}$ T m/A and $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m gives $c = 3.0 \times 10^8$ m/s. This matched mid-1800s measurements of the speed of light. Maxwell concluded that light is an EM wave, and that other wavelengths should exist. Hertz produced and detected radio waves in 1887, confirming the prediction.

What an EM wave looks like

At a snapshot in time, a plane EM wave travelling in the $+x$ direction has:

• $\vec{E}$ oscillating sinusoidally in (say) the $y$ direction,
• $\vec{B}$ oscillating in phase in the $z$ direction with $B_0 = E_0 / c$,
• both perpendicular to the direction of propagation (transverse wave),
• the wave carries energy and momentum but no rest mass.

The intensity (W m$^{-2}$) is proportional to $E_0^2$.

Worked example: comparing energies

A green photon ($\lambda = 550$ nm) and a UV photon ($\lambda = 200$ nm):

Green: $E = h c / \lambda = (6.626 \times 10^{-34})(3.0 \times 10^8) / (5.5 \times 10^{-7}) = 3.6 \times 10^{-19}$ J $= 2.3$ eV.

UV: $E = (6.626 \times 10^{-34})(3.0 \times 10^8) / (2.0 \times 10^{-7}) = 9.9 \times 10^{-19}$ J $= 6.2$ eV.

The UV photon carries roughly $2.75$ times the energy of the green one. This is enough to break a typical chemical bond ($\sim 4$ eV), which is why UV damages biological tissue.

Examples in context

Example 1. Parkes 64 m dish receiving 21 cm hydrogen-line radio waves. Neutral hydrogen in our galaxy emits at $\lambda = 21.1 \text{ cm}$. Using $c = f \lambda$ gives frequency $f = c / \lambda = 3.0 \times 10^8 / 0.211 = 1.42 \times 10^9 \text{ Hz} = 1.42 \text{ GHz}$. Each photon carries $E = h f = 6.626 \times 10^{-34} \times 1.42 \times 10^9 = 9.41 \times 10^{-25} \text{ J} = 5.87 \times 10^{-6} \text{ eV}$. This is far below the energy of visible photons ($\sim 2 \text{ eV}$), reflecting that radio waves probe low-energy transitions (the hyperfine spin-flip in hydrogen). The Parkes dish maps the galactic plane in this single line, tracing rotation curves that revealed dark matter.

Example 2. UV-C sterilisation at NSW Health pathology labs. A $254 \text{ nm}$ UV-C germicidal lamp emits photons of frequency $f = c/\lambda = 3.0 \times 10^8 / 2.54 \times 10^{-7} = 1.18 \times 10^{15} \text{ Hz}$ and energy $E = h f = 6.626 \times 10^{-34} \times 1.18 \times 10^{15} = 7.83 \times 10^{-19} \text{ J} = 4.89 \text{ eV}$. This energy exceeds the $\sim 4 \text{ eV}$ needed to dimerise adjacent thymine bases in bacterial DNA, killing the pathogen. Visible light at $\lambda = 500 \text{ nm}$ has $E = 2.48 \text{ eV}$, too low to break DNA bonds - which is why office lighting does not sterilise but UV-C cabinets do.

Try this

Q1. List the seven main bands of the electromagnetic spectrum in order of increasing frequency. [2 marks]

• Cue. Radio, microwave, infrared, visible, ultraviolet, X-ray, gamma. One mark if order is correct, second for all seven named.

Q2. A laser pointer emits red light at $\lambda = 650 \text{ nm}$. Calculate (a) the frequency and (b) the photon energy in eV. [3 marks]

• Cue. (a) $f = c/\lambda = 4.62 \times 10^{14} \text{ Hz}$. (b) $E = hf = 3.06 \times 10^{-19} \text{ J} = 1.91 \text{ eV}$.

Q3. Maxwell's equations predict EM waves travel at $c = 1 / \sqrt{\mu_0 \varepsilon_0}$ in vacuum. (a) Calculate $c$ from $\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$ and $\varepsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$. (b) Explain how this prediction identified light as an electromagnetic wave. (c) State one experimental verification of Maxwell's prediction. [2+2+2 marks]

• Cue. (a) $c = 1/\sqrt{4\pi \times 10^{-7} \times 8.85 \times 10^{-12}} = 3.00 \times 10^8 \text{ m/s}$. (b) Matched known speed of light. (c) Hertz (1888) generating radio waves and measuring $c$.