Advanced E&M Treatment of the Poynting Vector Applied To Hertzian Dipoles

 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]

The preceding is the most elementary form usually presented in the 2nd semester of Calculus physics.  But in this post I want to treat the more advanced (graduate) level form:

S = 1/m_o  [E X B]   =

 m_o / c  {[ P_o  w² / 4p ]  sin q/ r  cos w² (t - r/c)]}²    r^

^{From here we proceed by taking the time average, i.e.:}

^{<S> =}    (m_o / c  )P_o ² w⁴ / 32 p ²  sin ² q/ r²    r^

Then the total power radiated:      

P_total    = m_o P_o ² w⁴ / 32 p ²  c  ò ^2p  _o  df   ò ^p  _o  sin ² q  sin q dq

ò ^p  _o  sin ² q  sin q dq  =   ò ^p  _o  sin ³ q  dq  =  - cos q  + cos³ q / 3 ] ^p  _o

= - 1 - 1/3  -  (-1 + 1/3)  =  6/3  - 2/3   =  4/3

And:

  ò ^2p  _o  df  =    2 p

  Þ

P_total    =    

  m_o P_o ² w⁴ / 32 p ²  c (2 p)  (4/3) =  8p /3 [ m_o P_o ² w⁴ / 32 p ²  c]

P_total    =    m_o P_o ² w⁴ / 12 p  c

This amounts to the energy radiated from a Hertzian dipole, which is the largest of all types of dipole configuration.  

 Suggested Problem:

Find the total average power (P_{total av}) radiated by a Hertzian dipole over a closed sphere of radius r.   (Take  h_o as the impedance of free space, with I_o   the steady current and k  =  w /c  the wave number.