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Mathematics for Physics

79 Poynting Vector

A vector that is used to represent the directional energy flux density of an electromagnetic field, where energy flux density is the energy crossing a unit area per unit time.

The definition of the Poynting vector. The Poynting vector is built from the cross product of \(\mathbf{E}\) and \(\mathbf{B}\). This principle is used to assign a direction and magnitude to the flow of field energy.

The Poynting vector is

\[ \mathbf{S} = \dfrac{1}{\mu_{0}}\mathbf{E}\times\mathbf{B} \]

where

  • \(\mathbf{S}\) is the Poynting vector.
  • \(\mathbf{E}\) is the electric field.
  • \(\mathbf{B}\) is the magnetic field.
  • \(\mu_{0}\) is the permeability of free space.

Its flux as power. The flux of \(\mathbf{S}\) through a closed surface is the rate at which field energy leaves the enclosed volume. This principle is used to compute the power carried by a wave or drained from a circuit.

The outward energy current is

\[ P = \oint\mathbf{S}\cdot d\mathbf{a} \]

where

  • \(P\) is the outward power.
  • \(\mathbf{S}\) is the Poynting vector.
  • \(d\mathbf{a}\) is the outward vector area element.

The energy flow of a plane wave. In a plane electromagnetic wave \(\mathbf{S}\) points along the travel direction. This principle is used to identify the direction of energy transport with the direction of the wave.

The magnitude of the Poynting vector of a plane wave is

\[ S = \dfrac{1}{\mu_{0}c}E^{2} \]

where

  • \(S\) is the magnitude of \(\mathbf{S}\).
  • \(E\) is the electric field magnitude.
  • \(c\) is the speed of light.
  • \(\mu_{0}\) is the permeability of free space.