Poynting’s theorem

Poynting s theorem for the energy flow of plane waves in vacuo thus applies to the EM and EMS modes, but not to the S mode. Vector multiplication of Eqs. (52) and (53) by k, and combination with Eq. (49) and the result E C = 0, is easily shown [16,20] to result in a Poynting vector that is parallel with the group velocity C of Eq. (56). Later in Section VII.C.3 we shall return to Poynting s theorem in the case of axisymmetric photon wavepackets. 

Figure 8-1 Derivation of Poynting s theorem and the energy inequality.

Expression (8.92) represents a differential form of Poynting s theorem. 

An important property of the energy flow Ft out of domain V), containing sources j , is that its integral over a time period, Ftdt, is always non-negative independently of the type of the extraneous sources. To prove this we introduce a domain Oft Pi formed by the ball without domain (Figure 8-1). We can apply now Poynting s theorem (8.95) to the volume Oh Vi, taking into account that extraneous currents j = 0 in Or Vi ... 

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. ... 

Following the technique outlined above for the time domain case, we apply Poynting s theorem to the volume 0 V], taking into account that extraneous currents j = 0 in Or Vi ... 

We now can write Poynting s theorem (8.108) for the field E, H given within domain V ... 

By means of Poynting s " theorem, it can be shown that gi gi = g- Equation (15.7) holds for a dissymmetric isotropic medium. With Maxwell s equations (15.4), Eq. (15.7), neglecting the rotational term, become... 

Using Poynting s theorem, finally, after some mathematics, the extinction cross section Cext for the dipole is obtained as... 

According to the energy conservation law Poynting s theorem, the work done by the electric field is... 

According to Poynting s theorem, the extinction spectra, Pexi (o), the scattering spectra, Pscaipi) and the absorption spectra, Pabs oi), are given respectively by ... 

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