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Mathematics for Physics

73 Momentum of Light

The momentum carried by electromagnetic radiation that is used to count field momentum together with mechanical momentum so that total momentum is conserved.

The momentum density of a light wave. A plane electromagnetic wave carries momentum in the direction of travel. This principle is used to treat a light beam as a stream of momentum as well as energy.

The momentum density of a plane wave is

\[ \mathbf{g} = \dfrac{1}{c^{2}}\mathbf{S} \]

where

  • \(\mathbf{g}\) is the momentum density.
  • \(\mathbf{S}\) is the Poynting vector.
  • \(c\) is the speed of light.

The momentum of absorbed energy. The momentum delivered to a perfect absorber equals the absorbed energy divided by \(c\). This principle is used to compute the force of a light beam on a black surface.

The momentum of an absorbed energy \(U\) is

\[ p = \dfrac{U}{c} \]

where

  • \(p\) is the delivered momentum.
  • \(U\) is the absorbed energy.
  • \(c\) is the speed of light.

The radiation pressure on a reflector. A perfect reflector receives twice that momentum. This principle is used to compute radiation pressure on a mirror.

The radiation pressure on a perfect reflector is

\[ P_{\mathrm{rad}} = \dfrac{2S}{c} \]

where

  • \(P_{\mathrm{rad}}\) is the radiation pressure.
  • \(S\) is the magnitude of the incident Poynting vector.
  • \(c\) is the speed of light.