Electromagnetic Theory

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

$$\frac{\partial E_x}{\partial t} =-\frac{1}{\epsilon_0}\left(j_x + \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$$ (C.1)

$$\frac{\partial E_y}{\partial t} =-\frac{1}{\epsilon_0}\left(j_y + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$$ (C.2)

$$\frac{\partial E_z}{\partial t} =-\frac{1}{\epsilon_0}\left(j_z + \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$$ (C.3)

$$\frac{\partial H_x}{\partial t} =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$$ (C.4)

$$\frac{\partial H_y}{\partial t} = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$$ (C.5)

$$\frac{\partial H_z}{\partial t} =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right)$$ (C.6)

(Fitzpatrick 2008). Here, ${\bf E}$ is the electric field-strength, ${\bf H}$ the magnetic intensity, ${\bf j}$ the current density (i.e., the current per unit area), $\epsilon_0$ the electric permittivity of free space, and $\mu_0$ the magnetic permeability of free space.

For the case of a vacuum,

$${\bf j} = {\bf0}.$$ (C.7)

Hence, the previous equations simplify to give

$$\frac{\partial E_x}{\partial t} =-\frac{1}{\epsilon_0}\left(\frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$$ (C.8)

$$\frac{\partial E_y}{\partial t} = -\frac{1}{\epsilon_0}\left(\frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$$ (C.9)

$$\frac{\partial E_z}{\partial t} = -\frac{1}{\epsilon_0}\left(\frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$$ (C.10)

$$\frac{\partial H_x}{\partial t} =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$$ (C.11)

$$\frac{\partial H_y}{\partial t} = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$$ (C.12)

$$\frac{\partial H_z}{\partial t} =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right).$$ (C.13)

For the case of a dielectric medium,

$${\bf j} = \frac{\partial{\bf P}}{\partial t},$$ (C.14)

where ${\bf P}$ is the electric dipole moment per unit volume (Fitzpatrick 2008). Hence, we obtain

$$\frac{\partial E_x}{\partial t} =-\frac{1}{\epsilon_0}\left(\frac{\partial P_x}{\partial t} + \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$$ (C.15)

$$\frac{\partial E_y}{\partial t} =-\frac{1}{\epsilon_0}\left(\frac{\partial P_y}{\partial t} + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$$ (C.16)

$$\frac{\partial E_z}{\partial t} =-\frac{1}{\epsilon_0}\left(\frac{\partial P_z}{\partial t} + \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$$ (C.17)

$$\frac{\partial H_x}{\partial t} =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$$ (C.18)

$$\frac{\partial H_y}{\partial t} = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$$ (C.19)

$$\frac{\partial H_z}{\partial t} =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right).$$ (C.20)

Now, the electric displacement is defined

$${\bf P}= \epsilon_0\,{\bf E}+{\bf P}.$$ (C.21)

Moreover, in a linear dielectric medium,

$${\bf D} = \epsilon_0\,\epsilon\,{\bf E},$$ (C.22)

where $\epsilon$ is the relative dielectric constant (Fitzpatrick 2008). Thus, we get

$$\frac{\partial D_x}{\partial t} = \frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z},$$ (C.23)

$$\frac{\partial D_y}{\partial t} = \frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x},$$ (C.24)

$$\frac{\partial D_z}{\partial t} = \frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y},$$ (C.25)

$$\frac{\partial B_x}{\partial t} = \frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y},$$ (C.26)

$$\frac{\partial B_y}{\partial t} = \frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z},$$ (C.27)

$$\frac{\partial B_z}{\partial t} = \frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x},$$ (C.28)

where ${\bf B}=\mu_0\,{\bf H}$ is the magnetic field-strength. The previous three equations can also be written

$$\frac{\partial H_x}{\partial t} = v^{\,2}\left(\frac{\partial D_y}{\partial z}-\frac{\partial D_z}{\partial y}\right),$$ (C.29)

$$\frac{\partial H_y}{\partial t} =v^{\,2}\left(\frac{\partial D_z}{\partial x}- \frac{\partial D_x}{\partial z}\right),$$ (C.30)

$$\frac{\partial H_z}{\partial t} = v^{\,2}\left(\frac{\partial D_x}{\partial y}-\frac{\partial D_y}{\partial x}\right),$$ (C.31)

where $v=c/\sqrt{\epsilon}$ is the characteristic wave speed, and $c=1/\sqrt{\epsilon_0\,\mu_0}$ the speed of light in vacuum.

In an Ohmic conductor,

$${\bf j} = \sigma\,{\bf E},$$ (C.32)

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

$$\frac{\partial E_x}{\partial t} =-\frac{1}{\epsilon_0}\left(\sigma\,E_x+ \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$$ (C.33)

$$\frac{\partial E_y}{\partial t} =-\frac{1}{\epsilon_0}\left(\sigma\,E_y + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$$ (C.34)

$$\frac{\partial E_z}{\partial t} =-\frac{1}{\epsilon_0}\left(\sigma\,E_z+ \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$$ (C.35)

$$\frac{\partial H_x}{\partial t} =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$$ (C.36)

$$\frac{\partial H_y}{\partial t} = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$$ (C.37)

$$\frac{\partial H_z}{\partial t} =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right)$$ (C.38)

The energy flux associated with an electromagnetic wave has the components

$${\cal I}_x = E_y\,H_z-E_z\,H_y,$$ (C.39)

$${\cal I}_y = E_z\,H_x-E_x\,H_z,$$ (C.40)

$${\cal I}_z = E_x\,H_y-E_y\,H_x,$$ (C.41)

irrespective of the medium (Fitzpatrick 2008).

If $z=0$ 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

$$[E_x]_{z=0_-}^{z=0_+} =0,$$ (C.42)

$$[E_y]_{z=0_-}^{z=0_+} =0,$$ (C.43)

$$[D_z]_{z=0_-}^{z=0_+} =0,$$ (C.44)

$$[H_x]_{z=0_-}^{z=0_+} =0,$$ (C.45)

$$[H_y]_{z=0_-}^{z=0_+} =0,$$ (C.46)

$$[H_z]_{z=0_-}^{z=0_+} = 0$$ (C.47)

(Fitzpatrick 2008).

The equation of motion of a particle of mass $m$ and charge $q$ situated in electric and magnetic fields is

$$m\,\frac{d^{\,2}x}{dt^{\,2}} = q\left(E_x + B_z\,\frac{d y}{d t}-B_y\,\frac{d z}{d t}\right),$$ (C.48)

$$m\,\frac{d^{\,2}y}{dt^{\,2}} = q\left(E_y + B_x\,\frac{dz}{d t}-B_z\,\frac{d x}{d t}\right),$$ (C.49)

$$m\,\frac{d^{\,2}z}{dt^{\,2}} = q\left(E_z + B_y\,\frac{d x}{d t}-B_x\,\frac{d y}{d t}\right),$$ (C.50)

where $(x,\,y,\,z)$ are the particle's Cartesian coordinates, and ${\bf B}=\mu_0\,{\bf H}$ is the magnetic field-strength (Fitzpatrick 2008).