Physics · Lesson 27

Quantum ideas: photoelectric effect, energy levels, X-rays, wave–particle duality

JAMBWAECNECO

What the syllabus asks

JAMB: elementary modern physics: photoelectric effect, energy quantisation, X-rays, wave–particle paradox

WAEC: quantum ideas: black-body radiation and Planck's quantum hypothesis (E = hf), photoelectric effect, Einstein's photoelectric equation, work function, threshold frequency, stopping potential, photocells and their uses; models of the atom, energy levels and spectra (emission and absorption, line spectra of hydrogen), ionisation and excitation; X-rays: production, properties, uses and hazards;…

NECO: quantum ideas: black-body radiation and Planck's quantum hypothesis (E = hf), photoelectric effect, Einstein's photoelectric equation, work function, threshold frequency, stopping potential, photocells and their uses; models of the atom, energy levels and spectra (emission and absorption, line spectra of hydrogen), ionisation and excitation; X-rays: production, properties, uses and hazards;…

What you will learn

State Planck's quantum hypothesis and calculate photon energies.
Describe the photoelectric effect and explain why the wave theory cannot account for it.
Use Einstein's equation with work function, threshold frequency and stopping potential.
Explain energy levels, excitation, ionisation and emission and absorption line spectra.
Calculate the wavelength of radiation from energy-level transitions.
Describe the production, properties, uses and hazards of X-rays.
Explain wave–particle duality and calculate de Broglie wavelengths.

Key points

Planck: energy comes in quanta; photon energy E = hf = hc/λ; h = 6.63 × 10⁻³⁴ J s.
Intensity = number of photons; frequency = energy per photon; 1 eV = 1.6 × 10⁻¹⁹ J.
Photoelectric laws: threshold f₀, instant emission, current ∝ intensity, KEmax ∝ (f − f₀).
Einstein: hf = φ + ½mv²max; φ = hf₀; eV₀ = KEmax.
KEmax–f graph: straight line, gradient h, f-intercept f₀, KE-intercept −φ.
Photocells: alarms, automatic doors, counters, street lights, light meters.
Atoms have discrete levels; emitted/absorbed photon hf = E₂ − E₁; hydrogen Eₙ = −13.6/n² eV.
Line spectra are element fingerprints; emission (bright) and absorption (dark) lines.
X-rays: electrons stopped by tungsten; current → intensity, voltage → penetration; λmin = hc/eV; uses and lead shielding.
Wave–particle duality; de Broglie λ = h/mv; electron diffraction and electron microscopes.

Start here: Why "modern" physics was needed

By 1900, physicists believed light was entirely a wave. Waves explained reflection, refraction, interference and diffraction beautifully (the lesson “Waves”). But a few experiments refused to fit. The most important were:

Black-body radiation: the colour and brightness of glowing hot objects, like the red glow of a blacksmith's iron or the orange coil of an electric stove.
The photoelectric effect: light knocking electrons out of metals.
Line spectra: hot gases emit only certain sharp colours, not a continuous rainbow.

In 1900 Max Planck solved the first puzzle with a daring idea: energy is not emitted continuously, but in tiny separate packets called quanta.

Planck's quantum hypothesis: electromagnetic energy is emitted or absorbed in discrete packets (quanta). The energy of one quantum of frequency f is E = hf, where h = 6.63 × 10⁻³⁴ J s (Planck's constant). Since c = fλ, also E = hc/λ.

Einstein later called these packets photons. High-frequency radiation (violet, ultraviolet, X-rays) has high-energy photons; low-frequency radiation (red, infrared, radio) has low-energy photons. This explains why ultraviolet light from the sun can damage skin cells while far brighter red light does not.

Worked example 27.1

Find the energy of a photon of green light of wavelength 500 nm, in joules and in electron-volts.

Convert: 500 nm = 5.0 × 10⁻⁷ m.
E = hc/λ = (6.63 × 10⁻³⁴ × 3.0 × 10⁸)/(5.0 × 10⁻⁷) = 3.98 × 10⁻¹⁹ J.
In eV: 3.98 × 10⁻¹⁹ / 1.6 × 10⁻¹⁹ ≈ 2.49 eV.

3.98 × 10⁻¹⁹ J ≈ 2.5 eV

⚠️ Common mistake: forgetting to convert nm to m (1 nm = 10⁻⁹ m).

Try it yourself. Find the energy of a photon of frequency 6.0 × 10¹⁴ Hz in eV.

Show solution

E = 6.63 × 10⁻³⁴ × 6.0 × 10¹⁴ = 3.98 × 10⁻¹⁹ J = ≈ 2.5 eV.

Continue this lesson in the free app →

Practice questions

1. Find the energy of a photon of wavelength 400 nm, in eV.

Show worked answer

hc/λ = 4.97 × 10⁻¹⁹ J ≈ 3.1 eV.

2. A metal has φ = 3.0 eV. Find its threshold frequency.

Show worked answer

4.8 × 10⁻¹⁹/6.63 × 10⁻³⁴ ≈ 7.2 × 10¹⁴ Hz.

3. Light of 5.0 eV falls on a metal with φ = 2.0 eV. Find KEmax in J and the stopping potential.

Show worked answer

3.0 eV = 4.8 × 10⁻¹⁹ J; V₀ = 3.0 V.

4. An electron in hydrogen falls from n = 4 (−0.85 eV) to n = 2 (−3.4 eV). Find the photon energy.

Show worked answer

2.55 eV (blue-green, ≈ 486 nm).

5. An X-ray tube works at 30 kV. Find λmin.

Show worked answer

1.989 × 10⁻²⁵/(4.8 × 10⁻¹⁵) ≈ 4.1 × 10⁻¹¹ m.

Key words

Quantum: The smallest discrete amount of energy that can be emitted or absorbed.
Photon: A quantum (packet) of electromagnetic radiation, energy hf.
Planck's constant h: 6.63 × 10⁻³⁴ J s, linking photon energy and frequency.
Photoelectric effect: The emission of electrons from a metal surface when light of high enough frequency falls on it.
Photoelectron: An electron emitted in the photoelectric effect.
Work function φ: The minimum energy needed to remove an electron from the surface of a metal.
Threshold frequency f₀: The minimum frequency of light that causes photoemission, f₀ = φ/h.
Stopping potential: The reverse p.d. that just stops the fastest photoelectrons.
Electron-volt (eV): The energy gained by an electron accelerated through 1 V: 1.6 × 10⁻¹⁹ J.
Energy level: One of the allowed energies of an electron in an atom.
Excitation: Raising an electron to a higher energy level.
Ionisation energy: The energy needed to remove an electron completely from an atom in its ground state.

Learn Quantum ideas: photoelectric effect, energy levels, X-rays, wave–particle duality step by step, free.
The Cara's Tips app teaches every topic from the very basics, with quick checks, worked examples and exam practice. It works offline.
Open this lesson in the free app Download for Android