Electromagnetic Theory
The subset of Maxwell's equations that governs the propagation of electromagnetic waves can be written


(C.1) 


(C.2) 


(C.3) 


(C.4) 


(C.5) 


(C.6) 
(Fitzpatrick 2008). Here, is the electric fieldstrength, the magnetic intensity, the current density (i.e., the current per
unit area),
the electric permittivity of free space, and the magnetic permeability of free space.
For the case of a vacuum,

(C.7) 
Hence, the previous equations simplify to give


(C.8) 


(C.9) 


(C.10) 


(C.11) 
For the case of a dielectric medium,

(C.14) 
where is the electric dipole moment per unit volume (Fitzpatrick 2008). Hence, we obtain
Now, the electric displacement is defined

(C.21) 
Moreover, in a linear dielectric medium,

(C.22) 
where is the relative dielectric constant (Fitzpatrick 2008).
Thus, we get


(C.23) 


(C.24) 


(C.25) 


(C.26) 


(C.27) 


(C.28) 
where
is the magnetic fieldstrength.
The previous three equations can also be written
where
is the characteristic wave speed, and
the
speed of light in vacuum.
In an Ohmic conductor,

(C.32) 
where is the electrical conductivity (Fitzpatrick 2008). Thus, the equations governing
electromagnetic wave propagation in such a
medium become


(C.33) 


(C.34) 


(C.35) 


(C.36) 


(C.37) 


(C.38) 
The energy flux associated with an electromagnetic wave has the components
irrespective of the medium (Fitzpatrick 2008).
If is the interface between two different (nonmagnetic) media then the general matching conditions
for the components of the electric and magnetic fields across the interface are


(C.42) 


(C.43) 


(C.44) 


(C.45) 


(C.46) 


(C.47) 
(Fitzpatrick 2008).
The equation of motion of a particle of mass and charge situated in electric and magnetic fields is
where
are the particle's Cartesian coordinates, and
is the magnetic fieldstrength (Fitzpatrick 2008).