The Electromagnetic Spectrum
In 1864, James Clerk Maxwell at King's College London published a unified theory predicting electromagnetic waves travel at exactly $c = 2.998 \times 10^8$ m/s — then in 1887, Heinrich Hertz generated 3 GHz radio waves in his Karlsruhe laboratory and measured their 66 cm wavelength, confirming Maxwell's prediction to within 1.5%. Every region of the electromagnetic spectrum — from radio to gamma — obeys the same equations Maxwell wrote.
Practise this lesson
Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
Consider a radio wave and a gamma ray. A radio wave might have a wavelength of 100 m, while a gamma ray has a wavelength of $10^{-12}$ m.
Before reading on, answer:
- Which wave has the higher frequency? Explain your reasoning.
- Which wave carries more energy per photon? Why?
- Both waves travel at the same speed in vacuum. Does this mean they have the same effect on matter? Explain.
Warm-up — in the electromagnetic spectrum, which type of radiation has the shortest wavelength?
Know — The EM Spectrum
- Seven regions: radio, microwave, IR, visible, UV, X-ray, gamma
- All EM waves are transverse and travel at $c = 3.0\times10^8$ m/s in vacuum
- Relationship: $c = f\lambda$
Understand — Photon Energy
- Photon energy $E = hf = hc/\lambda$
- Higher frequency = higher energy per photon
- Why gamma rays are dangerous and radio waves are harmless
Can Do — Calculate & Compare
- Calculate frequency from wavelength and vice versa
- Calculate photon energy in joules and electronvolts
- Compare wavelength, frequency and energy across regions
Core Content
One phenomenon, seven regions
Tune a radio to an FM station and music reaches you through nothing but empty space. Stand outside in the dark and warmth from the sun still finds you. Undergo an X-ray and the image forms without the beam touching you. In each case, energy has arrived as an oscillating electric and magnetic field — a self-propagating electromagnetic wave requiring no medium, travelling at exactly $c = 3.00 \times 10^8$ m/s. All electromagnetic waves share three fundamental properties: they are transverse waves, they travel at the speed of light in vacuum, and they do not require a medium.
The electromagnetic spectrum is traditionally divided into seven regions, ordered by increasing frequency (and decreasing wavelength):
Figure 1 — The electromagnetic spectrum: all regions travel at speed $c$ in vacuum, differing only in wavelength and frequency
Radio waves ($\lambda \sim 10^3$ m): Used for communication — AM/FM radio, television, mobile phones. Lowest energy, harmless.
Microwaves ($\lambda \sim 10^{-2}$ m): Used in microwave ovens (heating water molecules), radar, and Wi-Fi. Can cause heating of tissue.
Infrared ($\lambda \sim 10^{-5}$ m): Felt as heat. Used in remote controls, thermal imaging, and night vision. Can cause burns at high intensity.
Visible light ($\lambda \sim 400$–$700$ nm): The only region our eyes can detect. Red has the longest wavelength ($\approx 700$ nm); violet the shortest ($\approx 400$ nm).
Ultraviolet ($\lambda \sim 10^{-8}$ m): Causes sunburn and skin damage. Used in sterilisation and forensics. Can damage DNA.
X-rays ($\lambda \sim 10^{-10}$ m): Pass through soft tissue but are absorbed by bone. Used in medical imaging and airport security. Ionising — dangerous in high doses.
Gamma rays ($\lambda \sim 10^{-12}$ m): Highest energy, produced by nuclear decay and cosmic events. Used in cancer treatment (radiotherapy). Highly ionising and dangerous.
Arrange the following in order of increasing photon energy: red light, blue light, infrared, ultraviolet, X-rays. Explain your reasoning using the relationship $E = hf$.
The electromagnetic spectrum has seven regions ordered by increasing frequency: radio, microwave, infrared, visible, UV, X-ray, gamma. All EM waves are transverse and travel at $c = 3.00 \times 10^8$ m/s in vacuum — no medium required. Higher frequency means shorter wavelength and greater photon energy ($E = hf$).
Pause — copy the highlighted definition into your book before moving on.
Which of the following correctly orders three regions of the EM spectrum from lowest to highest photon energy?
The universal constant that connects wavelength and frequency
We just saw that the EM spectrum spans seven regions all travelling at $c$. That raises a question: how do frequency and wavelength relate mathematically, and what determines a photon's energy? This card answers it → the wave equation $c = f\lambda$ and the photon energy formula $E = hf$.
All electromagnetic waves in vacuum obey the fundamental wave equation $c = f\lambda$, where $c = 3.00 \times 10^8$ m/s is the speed of light in vacuum, $f$ is frequency in hertz (Hz), and $\lambda$ is wavelength in metres (m). Since $c$ is constant, wavelength and frequency are inversely proportional: high frequency means short wavelength, and vice versa.
Each photon of electromagnetic radiation carries energy:
$c = f\lambda$ ($c = 3.00\times10^8$ m/s in vacuum)
$E = hf = \dfrac{hc}{\lambda}$ ($h = 6.63\times10^{-34}$ J·s)
$1\text{ eV} = 1.60\times10^{-19}\text{ J}$
These equations reveal why gamma rays are dangerous and radio waves are harmless: a gamma photon carries $\sim10^{15}$ times more energy than a radio photon, enough to ionise atoms and damage DNA strands.
When EM waves enter a medium such as glass or water, they slow down. The speed in a medium is $v = c/n$, where $n$ is the refractive index. The frequency stays the same, but the wavelength decreases: $\lambda_{\text{medium}} = \lambda_{\text{vacuum}}/n$.
Figure 2 — Wave parameters: wavelength is the distance between successive crests; amplitude is the maximum displacement from equilibrium
A microwave oven operates at 2.45 GHz. Calculate the wavelength of these microwaves. If these waves enter water ($n = 1.33$), what is their new wavelength and speed?
For all EM waves: $c = f\lambda$ (speed = frequency × wavelength, $c = 3.00 \times 10^8$ m/s). Photon energy $E = hf = hc/\lambda$ where $h = 6.63 \times 10^{-34}$ J·s. In a medium, speed decreases to $v = c/n$, wavelength decreases, but frequency remains unchanged.
Pause — write the highlighted wave equation and photon energy formula into your book before the check below.
A radio station broadcasts at 100 MHz. The wavelength of these waves is _____ m. (Use $c = 3.0 \times 10^8$ m/s)
From wavelength to energy across the spectrum
We just saw that $E = hf$ links frequency to photon energy. That raises a question: how do you apply this to concrete calculations involving wavelengths in different units or energies in electronvolts? This card answers it → a step-by-step approach to photon calculations across the spectrum.
Calculate the frequency and photon energy of red light with wavelength 650 nm. Express the energy in both joules and electronvolts.
$f = c/\lambda = 3.00\times10^8 / (650\times10^{-9}) = 4.62\times10^{14}$ Hz
$E = hf = 6.63\times10^{-34} \times 4.62\times10^{14} = 3.06\times10^{-19}$ J
$E = 3.06\times10^{-19} / 1.60\times10^{-19} = 1.91$ eV
An X-ray photon has energy 50 keV. Calculate its wavelength and frequency.
$E = 50\times10^3 \times 1.60\times10^{-19} = 8.0\times10^{-15}$ J
$f = E/h = 8.0\times10^{-15} / 6.63\times10^{-34} = 1.21\times10^{19}$ Hz
$\lambda = c/f = 3.00\times10^8 / 1.21\times10^{19} = 2.48\times10^{-11}$ m $= 0.025$ nm
A mobile phone tower emits radio waves at 900 MHz with total power 100 W. How many photons per second are emitted?
$E_{\text{photon}} = hf = 6.63\times10^{-34} \times 900\times10^6 = 5.97\times10^{-25}$ J
$N/s = P/E_{\text{photon}} = 100 / 5.97\times10^{-25} = 1.68\times10^{26}$ photons/s
Calculate the wavelength of a photon with energy 2.5 eV. In which region of the electromagnetic spectrum does this photon lie?
Photon calculation method: convert $\lambda$ to metres → $f = c/\lambda$ → $E = hf$ in joules → divide by $1.60 \times 10^{-19}$ for eV. Reverse: eV to J → $f = E/h$ → $\lambda = c/f$. Power emitted ÷ energy per photon = photons per second.
Add the highlighted calculation method to your notes before the check below.
A photon has energy $3.3\times10^{-19}$ J. What is its wavelength? (Use $hc = 1.99\times10^{-25}$ J·m)
Why photon energy determines safety, not wave speed
We just saw how to calculate photon energy from wavelength or frequency. That raises a question: since all EM waves travel at $c$, why do some harm living tissue while others do not? This card answers it → it is photon energy $E = hf$, not speed, that determines ionisation and biological effects.
Because all EM waves travel at the same speed, it might seem they should all have the same effect on matter. The key difference is photon energy — $E = hf$. High-frequency photons carry enough energy to ionise atoms (strip electrons away), damaging molecules including DNA. Low-frequency photons deposit only tiny amounts of energy — too little to cause ionisation.
Figure 3 — Ionising and non-ionising regions of the EM spectrum. UV, X-rays, and gamma rays carry sufficient photon energy to break chemical bonds
Key biological effects to know:
- UV: Damages DNA, causing sunburn and increasing skin cancer risk. Some UV is beneficial (vitamin D synthesis).
- X-rays: Penetrate soft tissue; absorbed by dense bone. Ionising — dose must be carefully managed in medical imaging.
- Gamma rays: Penetrate almost everything; produced in nuclear reactions. Used in cancer radiotherapy — targeted to tumour cells.
- Infrared: Causes heating (molecular vibrations), not ionisation. Prolonged exposure can burn skin.
- Radio/microwave: At normal intensities, these are non-ionising and harmless. Microwave ovens use resonant absorption by water molecules.
The EM spectrum spans many orders of magnitude. Common traps: confusing nanometres ($10^{-9}$ m) with micrometres ($10^{-6}$ m), or forgetting to convert MHz ($10^6$ Hz) to Hz before using $E = hf$. Always write your answer with the correct unit. Use $hc = 1.99\times10^{-25}$ J·m to speed up calculations when given wavelength.
Ionising radiation (UV, X-rays, gamma) carries sufficient photon energy to strip electrons from atoms and damage DNA — biological harm is determined by $E = hf$, not wave speed. Non-ionising radiation (radio, microwave, IR, visible) cannot directly ionise atoms. Medical uses include X-ray imaging, gamma radiotherapy, and UV sterilisation.
Add the highlighted principle to your notes before the check below.
All electromagnetic waves travel at different speeds in vacuum, which explains why gamma rays are more dangerous than radio waves.
When light enters glass, its frequency remains unchanged but its wavelength decreases.
Verify $c = f\lambda$ and compare photon energies across regions
- For radio waves ($\lambda = 1.0$ m): calculate the frequency and photon energy. Verify $c = f\lambda$ by hand.
- For visible light ($\lambda = 500$ nm): calculate how many times more energetic a visible photon is than a radio photon at 1.0 m.
- For gamma rays ($\lambda = 1.0\times10^{-12}$ m): calculate how many visible photons (500 nm) would equal the energy of one gamma photon.
- Explain why gamma rays are dangerous to living tissue while radio waves at the same power are harmless, using the photon energy equation.
Explain what changes and what stays constant when light enters a medium
A beam of red light ($\lambda = 700$ nm in air) enters a glass block ($n = 1.5$).
- Calculate the speed of light in the glass.
- Calculate the wavelength of the light inside the glass.
- Does the frequency change? Explain why or why not.
- A student argues: "Since the light slows down, it must have less energy per photon inside the glass." Is this correct? Explain.
In this lesson, three ideas lock together:
- All EM waves obey $c = f\lambda$ — one spectrum, seven regions differing only in wavelength/frequency.
- Photon energy $E = hf$ scales with frequency: gamma photons carry $\sim10^{15}\times$ more energy than radio photons.
- It is photon energy, not wave speed, that determines biological and physical effects of EM radiation.
How theory predicted that light is an electromagnetic wave
James Clerk Maxwell unified electricity and magnetism into one theory and showed, from that theory alone, that light is an electromagnetic wave.
In the 1860s, James Clerk Maxwell unified electricity and magnetism into a single theory of electromagnetism (Maxwell's equations). His equations predicted that a changing electric field produces a changing magnetic field, and vice versa — so the two propagate together as a self-sustaining electromagnetic wave through a vacuum, with no medium required.
Maxwell calculated the speed of these waves as $c = 1/\sqrt{\varepsilon_0\mu_0} \approx 3\times10^8$ m/s — determined only by the electric permittivity ($\varepsilon_0$) and magnetic permeability ($\mu_0$) of the vacuum. This matched the measured speed of light, leading to the conclusion that light is an electromagnetic wave.
Maxwell's work therefore unified optics with electricity and magnetism. His prediction was confirmed experimentally about 20 years later by Heinrich Hertz, who generated and detected radio waves travelling at the speed of light.
Copy the highlighted points (Maxwell's prediction of EM waves and $c = 1/\sqrt{\varepsilon_0\mu_0}$) into your book.
Maxwell calculated the speed of electromagnetic waves and found it equalled the measured speed of _____.
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ApplyBand 4(3 marks) 1. A laser pointer emits red light at 650 nm with a power of 3.0 mW. (a) Calculate the energy of each photon in joules. (b) Calculate the number of photons emitted per second. (c) If a blue laser emitted the same power at 450 nm, would it emit more or fewer photons per second? Explain.
1 mark: correct photon energy · 1 mark: correct photons/s · 1 mark: correct comparison with explanation
AnalyseBand 5(5 marks) 2. (a) Explain why all electromagnetic waves travel at the same speed in a vacuum. (b) When visible light enters glass ($n = 1.5$), describe what happens to its speed, wavelength, and frequency. (c) A student argues that since frequency is unchanged in glass, the photon energy is unchanged too. Evaluate this claim. (d) Explain using $E = hf$ why gamma rays are ionising but radio waves are not.
1 mark each: (a) Maxwell's equations prediction, (b) all three quantities correct, (c) evaluation, (d) energy comparison
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Multiple choice
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set drawn from the lesson bank.
Short Answer — Model Answers
Q1 (3 marks): (a) $f = c/\lambda = 3.00\times10^8 / 650\times10^{-9} = 4.62\times10^{14}$ Hz. $E = hf = 6.63\times10^{-34} \times 4.62\times10^{14} = 3.06\times10^{-19}$ J (1 mark). (b) $N/s = P/E = 3.0\times10^{-3} / 3.06\times10^{-19} = 9.8\times10^{15}$ photons/s (1 mark). (c) Blue photons at 450 nm have higher frequency and therefore more energy per photon ($E = hf$). The same power divided by a larger energy per photon gives fewer photons per second for blue light (1 mark).
Q2 (5 marks): (a) Maxwell's equations predicted self-propagating EM waves with speed $c = 1/\sqrt{\varepsilon_0\mu_0}$ — a constant determined by the properties of the vacuum, independent of frequency (1 mark). (b) Speed decreases: $v = c/n = 3.00\times10^8/1.5 = 2.0\times10^8$ m/s. Wavelength decreases: $\lambda' = \lambda/n = 700/1.5 = 467$ nm. Frequency is unchanged (1 mark). (c) The claim is correct: $E = hf$, frequency is unchanged in the medium, so photon energy is unchanged. What changes is the spatial extent of the wave (wavelength and speed), not the energy of individual photons (1 mark). (d) A gamma photon ($f \sim 3\times10^{20}$ Hz) has energy $E = hf \approx 2\times10^{-13}$ J, enough to ionise atoms (break chemical bonds). A radio photon ($f \sim 10^6$ Hz) has energy $\approx 7\times10^{-28}$ J — far too small to ionise any atom (1 mark).
At the start you were asked about Hertz's 1887 radio waves (wavelength 66 cm, frequency ~3 GHz, generated in his Karlsruhe laboratory) compared to a gamma ray ($\lambda \sim 10^{-12}$ m).
The answers: (1) The gamma ray has far higher frequency — since $c = f\lambda$ and both travel at $c$, the shorter-wavelength gamma ray must have higher frequency. (2) The gamma ray carries enormously more energy per photon: $E = hf$, so at $\lambda = 10^{-12}$ m, $E \approx 2\times10^{-13}$ J versus $\approx 2\times10^{-36}$ J for Hertz's radio wave — a factor of $\sim10^{23}$. (3) Same speed does NOT mean same effect — it is photon energy that determines biological impact. Hertz's confirmation of Maxwell's prediction established $c$ as a universal constant, and the huge energy ratio explains why gamma rays ionise tissue while radio waves are harmless.