understand and use the terms threshold frequency and threshold wavelength

22.1 Energy and Momentum of a Photon

Learning outcome

Students will be able to:

  • state the photon energy and momentum formulas,

  • convert photon energies between joules and electron‑volts,

  • define threshold frequency (ν0) and threshold wavelength (λ0) for the photo‑electric effect,

  • use these concepts in quantitative problems, and

  • relate photon properties to radiation pressure.

Theory

1. Photon energy

A photon is a mass‑less quantum of electromagnetic radiation. Its energy is directly proportional to its frequency:

E = h ν

  • h = Planck’s constant = 6.626 × 10⁻³⁴ J s

  • ν = frequency (Hz)

  • Using the wave relation c = λ ν, the energy can also be written as E = hc/λ.

2. Photon momentum

Even though a photon has no rest mass, it carries linear momentum:

p = h/λ = E/c

  • p = momentum (kg m s⁻¹)

  • λ = wavelength (m)

  • c = speed of light = 2.998 × 10⁸ m s⁻¹

3. Energy unit – electron‑volt (eV)

1 eV = 1.602 × 10⁻¹⁹ J. Converting between joules and eV is useful when dealing with atomic‑scale processes.

4. Threshold frequency and threshold wavelength

In the photo‑electric effect a photon can liberate an electron from a metal only if its energy exceeds the metal’s work function (ϕ). The minimum frequency required is called the threshold frequency (ν0); the corresponding wavelength is the threshold wavelength (λ0).

ν0 = ϕ / h

λ0 = c / ν0 = hc / ϕ

  • If ν < ν0 (or λ > λ0) no electrons are emitted, regardless of light intensity.

  • ν > ν0 (λ < λ0) results in electron emission with kinetic energy Kmax = h(ν – ν0) (Einstein’s photo‑electric equation).

5. Radiation pressure

When photons are absorbed or reflected they transfer momentum to a surface, producing a pressure:

P = I / c (absorbing surface)

P = 2I / c (reflecting surface)

  • I = intensity of the light (W m⁻²).

  • These relations follow directly from p = E/c.

Evidence box – Photo‑electric effect

ObservationInterpretation
Current appears instantly when light shines on a metal.Energy is delivered in discrete packets (photons) rather than by a wave‑like heating process.
No current if the light frequency is below a certain value, no matter how intense.A minimum photon energy (the work function) is required → defines ν0 and λ0.
Maximum kinetic energy of emitted electrons increases linearly with frequency.Kmax = h(ν – ν0) – quantitative confirmation of the photon model.

Worked example – Threshold frequency

Problem: The work function of sodium is 2.28 eV. Calculate the threshold frequency (ν0) and threshold wavelength (λ0) for sodium.

Solution:

  1. Convert the work function to joules:

    ϕ = 2.28 eV × 1.602 × 10⁻¹⁹ J eV⁻¹ = 3.65 × 10⁻¹⁹ J.

  2. Threshold frequency:

    ν0 = ϕ / h = (3.65 × 10⁻¹⁹ J) / (6.626 × 10⁻³⁴ J s) ≈ 5.5 × 10¹⁴ Hz.

  3. Threshold wavelength:

    λ0 = c / ν0 = (2.998 × 10⁸ m s⁻¹) / (5.5 × 10¹⁴ Hz) ≈ 5.5 × 10⁻⁷ m = 550 nm.

Thus visible light with wavelength shorter than 550 nm (e.g., blue or ultraviolet) can cause photo‑emission from sodium, whereas longer wavelengths cannot.

Table – Typical photon energies

Regionλ (nm)E (J)E (eV)
Radio10⁶ – 10⁹10⁻²⁸ – 10⁻²⁴10⁻⁹ – 10⁻⁵
Microwave1 mm – 1 cm10⁻²⁴ – 10⁻²²10⁻⁵ – 10⁻³
Infrared700 – 10⁴10⁻²² – 10⁻²⁰10⁻³ – 0.1
Visible400 – 7002.8 × 10⁻¹⁹ – 5.0 × 10⁻¹⁹1.8 – 3.1
Ultraviolet10 – 4005 × 10⁻¹⁹ – 2 × 10⁻¹⁷3 – 124
X‑ray0.01 – 102 × 10⁻¹⁶ – 2 × 10⁻¹⁴124 – 12 400

Connections to other syllabus topics

  • 22.2 Photo‑electric effect – uses the threshold frequency concept and the equation Kmax = h(ν – ν0).

  • 22.3 Wave‑particle duality – the simultaneous wave (λ, ν) and particle (E, p) descriptions of light.

  • 22.4 Quantum radiation – Compton scattering and line spectra rely on photon momentum conservation.

Practice questions

  1. A metal has a work function of 4.5 eV. Calculate its threshold wavelength.

  2. Find the momentum of a photon of wavelength 600 nm. Express your answer in kg m s⁻¹.

  3. If a laser of intensity 0.8 W m⁻² is completely absorbed by a surface, what is the radiation pressure exerted on the surface?

  4. Show that for a photon p = E/c using the relations E = hν and p = h/λ.