Poyntings theorem in the frequency domain

The energy flow of the electromagnetic field in this case can be calculated using the complex Poynting vector P (Stratton, 1941), introduced by the following formula  

The real part of the complex Poynting vector P may also be interpreted as the intensity of EM energy flow at a given point, that is the energy per second crossing a unit area whose normal is oriented in the direction of the vector E x H.  

In view of the Gauss formula, expression (8.105) can be cast in the form 

We can obtain Poynting s theorem by taking the scalar product of the second equation for the total field (8.103) with H and the complex conjugate of the first equation (8.102) with E, and subtracting one from the other  

This is Poynting s theorem in the frequency domain (Stratton, 1941). It says that the energy flow outside the domain V] is formed by the balance between the energy dissipated in heat per unit volume per second, and the energy introduced through tlie extraneous currents j.  

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