Right to the Poynting, attenuation and other amenities
The energy density (i.e. energy per unit volume) in electrodynamics is demonstrated in Feynman Ch. 27 to be closely related to the energy density as calculated for the static cases. The electrostatic calculation is developed from the energy expression for a system of discrete charges (Fey. 8.3):
(1)
which becomes, for continuous charge distribution:
(2)
where the last step is because the potential decrease as 1/R and its gradient as 1/R2 while the surface only increases as R2 .
The electrodynamic energy desity is:
(3)
If N is the number of charges per unit volume then the dissipated power in a unit volume, i.e. the rate of the electric field doing work on unit volume of matter is:
(4)
For power definitions in harmonic regime (which explain the extra 1/2 factors below) see the dedicated session. In sinusoidal steady-state conditions, the time-averaged stored electric and magnetic energy in a volume V are, in general and in the particular case of Lossless+Isotropic+Homogeneaous+Linear (LLIHL):
(5)
Then, once the electric source current density JS and the magnetic source current density MS are assigned the total electic current density is:
(6)
and from the dedicated section the Maxwell equations are written using Pozar's notation as:
(7)
where the source electric and magnetic are much like the "correnti impresse" in the Italian notations. Then by some algebraic/vector manipulation and allowing complex ε and µ to take losses into account, it is possible to derive the Poyinting's theorem that essentially describes the rate at which the EM field energy moves around:
(8)
where the time-average power delivered by the sources and flowing out of S is the real part of Ps, Po , and dissipation includes conduction, electric and magnetic loss. The vector S= ExH*is the Poynting's vector. In Pozar 1.7 it is shown that the time-average power flow is conserved at the interface between two media even (at z=0 which constitutes the two media interface) if the second one is lossy. Indeed for normal incidence of a plane wave the complex power and the time-average power are both conserved at z=0 and the real power power flow for z<0 can be decomposed in its incident and reflected waves components such that P-=Pi + Pr .
As the wave propagates into the lossy conductor medium, power is dissipated through an e-2α attenuation factor, therefore the average power dissipated in a 1m2 cross-sectional volume of the conductor is evaluated from (8) and from the expression for the electric volume current density i.e. the electric current density in a 1m2 cross-sectional volume (A/m2):
(9)
where Et stands for the transmitted field in the second medium. Thus from (8) and assuming a good conductor approximation ():
(10)
and this corresponds to the time-average power transmitted into the conductor per unit area (Collin pg. 55) i.e. the real power entering the lossy medium through a 1m2 cross-section:
(11)
If the conductor is perfect σà∞, then αà∞ , ηà0, Tà0, Γà-1 so the fields decay infinitely fast and are zero in the perfect conductor that "shorts out" the electric field at z=0 . As observed by Ramos pg 148, the tangential component of the magnetic field is zero inside the conductor (as the normal component) but it is not in general zero just outside. Indeed the magnetic field at z=0 is y(2/η0)E0 which is not 0 and follows (8) in the Maxwell Equations section:
(12)
The fact that in the limit of infinite conductivity the volume current density per unit width, i.e. the current density in a 1m2 cross-sectional volume (A/m2), reduces to the infinitely thin sheet of surface current (A/m) can be seen from the general expression for the current density through the 1m2 cross-sectional volume in (9). Indeed the total current in the x direction (because J is directed along x) per unit width is:
(13)
and vectors can be schematically plotted as in Fig.1.
Fig. (1)
Therefore in the limit of infinite conductivity:
(14)
which coincides with (12). Also it can be noticed that the Poynting vector in the same semispace is purely imaginary indicating that no real power is delivered to the perfect conductor.
(10) i.e. the calculation of power dissipated in the conductor from the current density in a 1m2 cross-sectional volume (A/m2) can be explicitly rewritten in the σà∞ limit, as well, leading to an expression that depends on the conductor surface resistance per unit length (Ramos pg 155):
(15)
Finally the dissipated power can also be assessed through direct calculation of the Poynting's vector at the conductor surface because all power entering at z=0 is dissipated, and again featuring a dependence on the surface resistance. From (11)
(16)
If the exponentially decaying volume (current in a 1m2 cross-sectional volume) current is replaced by a uniform volume current density extending within a δS length:
(17)
then the total current flow is the same as in (12) and the dissipated power is:
(18)
where Ht indicates the tangential component of the magnetic field and S is 1m2. This shows that loss can be computed in terms of surface resistance and surface current density or tangential magnetic field only, in the approximation of good (and thick) conductor.
The equations above are suitable to determine the attenuation α per unit length (i.e. when z=1m) in some practical transmission line configurations. The relationship between linear and logarithmic expression for the attenuation is:
(19)
Notice that the attenuation in dB for any other length can be simply scaled because any factor multiplied to α in the exponential in the second equation can be taken out of the exponential. For example α (dB/cm) implies evaluating the log[exp(2α*10-2)], which is simply 10-2*20log[exp(α)]=α (dB/m)/100 , as expected