Cambridge A‑Level Physics 9702 – Energy and Momentum of a Photon
Energy and Momentum of a Photon
Learning Objective
Understand that electromagnetic radiation possesses a particulate (photon) nature and be able to calculate the energy and momentum associated with a photon.
Key Concepts
A photon is a quantum of electromagnetic radiation.
Energy of a photon is directly proportional to its frequency.
Even though photons are mass‑less, they carry momentum.
The particulate description explains phenomena such as the photoelectric effect, Compton scattering and radiation pressure.
Energy of a Photon
The energy \$E\$ of a photon is given by Planck’s relation:
\$E = h\nu = \frac{hc}{\lambda}\$
where
\$h = 6.626\times10^{-34}\ \text{J·s}\$ is Planck’s constant,
\$\nu\$ is the frequency of the radiation,
\$c = 3.00\times10^{8}\ \text{m·s}^{-1}\$ is the speed of light in vacuum,
\$\lambda\$ is the wavelength.
Momentum of a Photon
From the relativistic energy‑momentum relation \$E^{2}=p^{2}c^{2}+m{0}^{2}c^{4}\$ and noting that the rest mass \$m{0}=0\$ for a photon, we obtain:
\$p = \frac{E}{c} = \frac{h}{\lambda}\$
Thus the momentum \$p\$ is inversely proportional to the wavelength.
Derivation from Relativistic Principles (Optional)
Starting with \$E = \gamma m{0}c^{2}\$ and \$p = \gamma m{0}v\$, for \$m_{0}=0\$ the Lorentz factor \$\gamma\$ becomes infinite, but the ratio \$E/p\$ remains finite and equals \$c\$. This leads directly to \$p = E/c\$ for a mass‑less particle.
Experimental Evidence for the Particulate Nature
Photoelectric Effect – Electrons are emitted from a metal only if the incident light has a frequency above a threshold, regardless of intensity. The kinetic energy of the emitted electrons follows \$K_{\max}=h\nu-\phi\$, where \$\phi\$ is the work function.
Compton Scattering – X‑rays scattered from electrons show a wavelength shift \$\Delta\lambda = \frac{h}{m_{e}c}(1-\cos\theta)\$, consistent with photon momentum transfer.
Radiation Pressure – Light exerts a measurable pressure \$P = \frac{I}{c}\$ (or \$2I/c\$ for perfect reflection) on surfaces, explained by momentum transfer from photons.
Suggested diagram: Schematic of the photoelectric effect showing incident photons, metal surface, and emitted electrons.
Sample Calculations
Table 1 illustrates the energy and momentum of photons at selected wavelengths.
Electromagnetic radiation can be described as a stream of photons, each carrying a quantised amount of energy \$E = h\nu\$ and a corresponding momentum \$p = h/\lambda\$. This particulate description successfully explains phenomena that wave theory alone cannot, confirming the dual nature of light.