Parsing the Poynting Vector

 The energy carried by an electromagnetic (E-M) wave with field intensities E, B is given by the Poynting vector, S:    

S = 1/m_o  [E X B]

where  m_o denotes the magnetic permeability, or m_o = 4 π x 10^-7  H/m

Physically, the Poynting vector is the rate at which energy flows through a unit surface perpendicular to the flow. We already saw one form based on working out the solar flux per unit area. 

This was given as: P = E _y  H _z / c  =   1300 Wm^-2 .

Recall now, for any two vectors A, B

A X B = A B sin Θ

where Θ is the angle between them. If then A is perpendicular to B, then sin (90) = 1 so that:

A B sin Θ = AB

By the same token, taking the vectors E, B perpendicular to each other, we find:   S = EB / m_o 

which has units of Wm^-2

Recall  E / B = c, so:

B = E /c

Then: S = EH = EB / m_o  = (E/c) E/ m_o  =   E ² / c m_o 

A depiction of an EM wave in terms of the Poynting vector S and E, H is shown below:

E-M Wave with Poynting vector (S)

Where:   B =  m_o H
or: S = (c /m_o) B

The "time average" is also of interest and entails taking the time average of the function:   cos ²(kx - wt)  

Which yields:

T _av {cos² (kx - wt) } = ½

The average value of S (or Intensity) can then be obtained from the maximum vector amplitudes, viz.

I = S _av = E _max B _max / 2 m_o

or:

S _av = E _max ²/ 2 m_o c  = cB _max ² / 2 m_o

Note that m_o c is a very important quantity known as the "impedance of free space", or:

m_o c  =    Öm_o  / Ö ε_o   

  = 377 ohms

The respective contribution of the two field energies (associated with the electric intensity, E, and magnetic intensity B) can easily be shown to be:

U _E = ½ ε_o  E²

U _m= ½ (B²/ m_o)

In a given volume the energy is equally shared by the two fields such that:

U _E = U _m = ½ ε_o E² = (B² /2 m_o)

The total, instantaneous energy density of the fields is then:

U = U _E + U _m = 2(½ ε_o E²) = ε_o  E² = B² /m_o

Averaged over one or more cycles  this leads to the total average energy per unit volume:

U _av = [ε_o  E² ] _av =

½ ε_o E _max ² = B _max ² /2 m_o

The intensity of an EM wave is then:

I = S _av = c U _av = PA

where P denotes the radiation pressure, or P = S/c (for complete absorption) or P = 2S/c for complete reflection of the wave.

(In direct sunlight, one finds P _R = 5 x 10^-6  N/m²)

Suggested Problem:

 A radio wave transmits 25 W/m ² of power per unit area. A plane surface of area 2.4 m x 0.7 m is perpendicular to the direction of propagation of the wave. Calculate the radiation pressure  P _R  on the surface if it is assumed to be a perfect absorber.