# 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 field-strength, 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)

 (C.12) (C.13)

For the case of a dielectric medium,

 (C.14)

where is the electric dipole moment per unit volume (Fitzpatrick 2008). Hence, we obtain

 (C.15) (C.16) (C.17) (C.18) (C.19) (C.20)

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 field-strength. The previous three equations can also be written

 (C.29) (C.30) (C.31)

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

 (C.39) (C.40) (C.41)

irrespective of the medium (Fitzpatrick 2008).

If is the interface between two different (non-magnetic) 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

 (C.48) (C.49) (C.50)

where are the particle's Cartesian coordinates, and is the magnetic field-strength (Fitzpatrick 2008).