Photons: (c) energy of a photon; E = hf and E = hc/λ

The idea

A photon is a quantum of electromagnetic radiation, and its energy depends only on frequency: E = hf, where h is the Planck constant (6.63 × 10⁻³⁴ J s) and f is frequency in hertz. Because c = fλ, substituting f = c/λ gives E = hc/λ, useful when wavelength is quoted. For example, red light of wavelength 6.5 × 10⁻⁷ m has E = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) ÷ (6.5 × 10⁻⁷) ≈ 3.1 × 10⁻¹⁹ J. Note that shorter wavelength means higher photon energy, and doubling frequency doubles energy. In MCQs, check whether the question gives f or λ, keep units consistent, and watch powers of ten.

What earns the marks

• States that photon energy is directly proportional to frequency, E = hf, with h = 6.63 × 10⁻³⁴ J s.
• Uses c = fλ to derive or apply E = hc/λ when wavelength rather than frequency is given.
• Recognises that shorter wavelength corresponds to higher photon energy for the same radiation speed c.
• Calculates correctly with powers of ten, for example E = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) ÷ (6.5 × 10⁻⁷) ≈ 3.1 × 10⁻¹⁹ J.
• Distinguishes photon energy from intensity: intensity depends on the number of photons per second, not on hf alone.

Common mistakes and how to put them right

• Using E = hf with wavelength substituted for f.

  Put it right: First finding f = c/λ, then applying E = hf.

• Forgetting to convert nanometres to metres.

  Put it right: 500 nm must be written as 5.00 × 10⁻⁷ m before substitution.

• Treating photon energy as dependent on intensity.

  Put it right: Intensity changes the number of photons, while each photon's energy depends only on frequency.

Exam tips

• Write the equation you will use before substituting numbers, so the examiner can follow your method.
• Convert all wavelengths to metres and keep values in standard form to reduce power-of-ten slips.
• Sanity-check the answer: visible photons have energies of order 10⁻¹⁹ J, so a result near 10⁻¹⁵ J signals an error.