Poynting Vector

Poynting Vector: Electromagnetic waves, like any other wave, can transport energy. The power through a unit area in a direction normal to the area is given by Poynting vector , given by

As and form a right handed triad, the direction of is along the direction of propagation. In SI units ismeasured in watt/m ²
The magnitude of for the electromagnetic wave travelling in vacuum is given by

where we have used the relationship between

and

in free space. For harmonic waves, we have

The average power transmitted per unit area, defined as the intensity is given by substituting the value 1/2 for the average of the square of sine or cosine function

Radiation Pressure: We have seen that electric field, as well as magnetic field, store energy. The energy density for the electric field was seen to be

and that for the magnetic field was found to be

. For the electromagnetic waves, where

, the total energy density is

where we have used

In addition to carrying energy, electromagnetic waves carry momentum as well. The relationship between energy ( U) and momentum ( P) is given by relatistic relation for a massless photons as

. Since the energy density of the electromagnetic waves is given by

, the momentum density, i.e. momentum per unit volume is

Since the direction of momentum must be along the direction of propagation of the wave, the above can be converted to a vector equation

If an electromagnetic wave strikes a surface, it will thus exert a pressure. Consider the case of a beam falling normally on a surface of area A which absorbs the wave. The force exerted on the surface is equal to the rate of change of momentum of the wave. The momentum change per unit time is given by the momentum contained within a volume

. The pressure, obtained by dividing the force by A is thus given by

which is exactly equal to the energy density u

If on the other hand, the surface reflects the wave, the pressure would be twice the above value.

The above is true for waves at normal incidence. If the radiation is diffuse, i.e., if it strikes the wall from all directions, it essentially consists of plane waves travelling in all directions. If the radiation is isotropic, the intensity of the wave is the same in all directions. The contribution to the pressure comes from those waves which are travelling in a direction which has a component along the normal to the surface. Thus on an average a third of the radiation is responsible for pressure. The pressure for an absorbing surface is u/3 while that for a reflecting surface is 2u/3

The existence of radiaton pressure can be verified experimentally. The curvature of a comet's tail is attributed to the radiation pressure exerted on the comet by solar radiation.