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Mathematics for Physics

48 Electromagnetic Momentum

The momentum carried by electromagnetic fields that is used to account for mechanical momentum changes when fields interact with matter.

The electromagnetic momentum density. The electromagnetic momentum density is proportional to \(\mathbf{E}\times\mathbf{B}\). This principle is used to assign a local momentum to the fields.

The electromagnetic momentum density is

\[ \mathbf{g} = \epsilon_{0}\mathbf{E}\times\mathbf{B} = \dfrac{1}{c^{2}}\mathbf{S} \]

where

  • \(\mathbf{g}\) is the field momentum density.
  • \(\mathbf{E}\) is the electric field.
  • \(\mathbf{B}\) is the magnetic field.
  • \(\mathbf{S}\) is the Poynting vector.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(c\) is the speed of light.

The total field momentum. The total field momentum is the integral of \(\mathbf{g}\) over space. This principle is used to include field momentum in the conservation law for an isolated system.

The total field momentum is

\[ \mathbf{P}_{\mathrm{field}} = \displaystyle\int\mathbf{g}\,d\tau \]

where

  • \(\mathbf{P}_{\mathrm{field}}\) is the total electromagnetic momentum.
  • \(\mathbf{g}\) is the momentum density.
  • \(d\tau\) is the volume element.

Radiation pressure. Radiation pressure is the momentum delivered per unit time per unit area. This principle is used to compute the force of a wave on an absorber or reflector.

The radiation pressure on a perfect absorber is

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

where

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