Electromagnetic Theory
The subset of Maxwell's equations that governs the propagation of electromagnetic waves can be written
![$\displaystyle \frac{\partial E_x}{\partial t}$](img1647.png) |
![$\displaystyle =-\frac{1}{\epsilon_0}\left(j_x + \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$](img4442.png) |
(C.1) |
![$\displaystyle \frac{\partial E_y}{\partial t}$](img2658.png) |
![$\displaystyle =-\frac{1}{\epsilon_0}\left(j_y + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$](img4443.png) |
(C.2) |
![$\displaystyle \frac{\partial E_z}{\partial t}$](img3220.png) |
![$\displaystyle =-\frac{1}{\epsilon_0}\left(j_z + \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$](img4444.png) |
(C.3) |
![$\displaystyle \frac{\partial H_x}{\partial t}$](img2000.png) |
![$\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$](img3217.png) |
(C.4) |
![$\displaystyle \frac{\partial H_y}{\partial t}$](img1649.png) |
![$\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$](img3218.png) |
(C.5) |
![$\displaystyle \frac{\partial H_z}{\partial t}$](img2002.png) |
![$\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right)$](img4445.png) |
(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,
![$\displaystyle {\bf j} = {\bf0}.$](img4446.png) |
(C.7) |
Hence, the previous equations simplify to give
![$\displaystyle \frac{\partial E_x}{\partial t}$](img1647.png) |
![$\displaystyle =-\frac{1}{\epsilon_0}\left(\frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$](img3205.png) |
(C.8) |
![$\displaystyle \frac{\partial E_y}{\partial t}$](img2658.png) |
![$\displaystyle = -\frac{1}{\epsilon_0}\left(\frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$](img3206.png) |
(C.9) |
![$\displaystyle \frac{\partial E_z}{\partial t}$](img3220.png) |
![$\displaystyle = -\frac{1}{\epsilon_0}\left(\frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$](img3221.png) |
(C.10) |
![$\displaystyle \frac{\partial H_x}{\partial t}$](img2000.png) |
![$\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$](img3217.png) |
(C.11) |
For the case of a dielectric medium,
![$\displaystyle {\bf j} = \frac{\partial{\bf P}}{\partial t},$](img4448.png) |
(C.14) |
where
is the electric dipole moment per unit volume (Fitzpatrick 2008). Hence, we obtain
Now, the electric displacement is defined
![$\displaystyle {\bf P}= \epsilon_0\,{\bf E}+{\bf P}.$](img4452.png) |
(C.21) |
Moreover, in a linear dielectric medium,
![$\displaystyle {\bf D} = \epsilon_0\,\epsilon\,{\bf E},$](img4453.png) |
(C.22) |
where
is the relative dielectric constant (Fitzpatrick 2008).
Thus, we get
![$\displaystyle \frac{\partial D_x}{\partial t}$](img1959.png) |
![$\displaystyle = \frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z},$](img2115.png) |
(C.23) |
![$\displaystyle \frac{\partial D_y}{\partial t}$](img2004.png) |
![$\displaystyle = \frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x},$](img2116.png) |
(C.24) |
![$\displaystyle \frac{\partial D_z}{\partial t}$](img1961.png) |
![$\displaystyle = \frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y},$](img2117.png) |
(C.25) |
![$\displaystyle \frac{\partial B_x}{\partial t}$](img2118.png) |
![$\displaystyle = \frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y},$](img2119.png) |
(C.26) |
![$\displaystyle \frac{\partial B_y}{\partial t}$](img2120.png) |
![$\displaystyle = \frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z},$](img2121.png) |
(C.27) |
![$\displaystyle \frac{\partial B_z}{\partial t}$](img2122.png) |
![$\displaystyle = \frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x},$](img2123.png) |
(C.28) |
where
is the magnetic field-strength.
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,
![$\displaystyle {\bf j} = \sigma\,{\bf E},$](img2891.png) |
(C.32) |
where
is the electrical conductivity (Fitzpatrick 2008). Thus, the equations governing
electromagnetic wave propagation in such a
medium become
![$\displaystyle \frac{\partial E_x}{\partial t}$](img1647.png) |
![$\displaystyle =-\frac{1}{\epsilon_0}\left(\sigma\,E_x+ \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$](img4457.png) |
(C.33) |
![$\displaystyle \frac{\partial E_y}{\partial t}$](img2658.png) |
![$\displaystyle =-\frac{1}{\epsilon_0}\left(\sigma\,E_y + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$](img4458.png) |
(C.34) |
![$\displaystyle \frac{\partial E_z}{\partial t}$](img3220.png) |
![$\displaystyle =-\frac{1}{\epsilon_0}\left(\sigma\,E_z+ \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$](img4459.png) |
(C.35) |
![$\displaystyle \frac{\partial H_x}{\partial t}$](img2000.png) |
![$\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$](img3217.png) |
(C.36) |
![$\displaystyle \frac{\partial H_y}{\partial t}$](img1649.png) |
![$\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$](img3218.png) |
(C.37) |
![$\displaystyle \frac{\partial H_z}{\partial t}$](img2002.png) |
![$\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right)$](img4445.png) |
(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 (non-magnetic) media then the general matching conditions
for the components of the electric and magnetic fields across the interface are
![$\displaystyle [E_x]_{z=0_-}^{z=0_+}$](img4464.png) |
![$\displaystyle =0,$](img437.png) |
(C.42) |
![$\displaystyle [E_y]_{z=0_-}^{z=0_+}$](img4465.png) |
![$\displaystyle =0,$](img437.png) |
(C.43) |
![$\displaystyle [D_z]_{z=0_-}^{z=0_+}$](img4466.png) |
![$\displaystyle =0,$](img437.png) |
(C.44) |
![$\displaystyle [H_x]_{z=0_-}^{z=0_+}$](img4467.png) |
![$\displaystyle =0,$](img437.png) |
(C.45) |
![$\displaystyle [H_y]_{z=0_-}^{z=0_+}$](img4468.png) |
![$\displaystyle =0,$](img437.png) |
(C.46) |
![$\displaystyle [H_z]_{z=0_-}^{z=0_+}$](img4469.png) |
![$\displaystyle = 0$](img4470.png) |
(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 field-strength (Fitzpatrick 2008).