next up previous contents
Next: Bibliography Up: Oscillations and Waves Previous: Trigonometric Identities   Contents


Electromagnetic Theory

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

$\displaystyle \frac{\partial E_x}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(j_x + \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$ (C.1)
$\displaystyle \frac{\partial E_y}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(j_y + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$ (C.2)
$\displaystyle \frac{\partial E_z}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(j_z + \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$ (C.3)
$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$ (C.4)
$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$ (C.5)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right)$ (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,

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

Hence, the previous equations simplify to give

$\displaystyle \frac{\partial E_x}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$ (C.8)
$\displaystyle \frac{\partial E_y}{\partial t}$ $\displaystyle = -\frac{1}{\epsilon_0}\left(\frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$ (C.9)
$\displaystyle \frac{\partial E_z}{\partial t}$ $\displaystyle = -\frac{1}{\epsilon_0}\left(\frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$ (C.10)
$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$ (C.11)

$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$ (C.12)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right).$ (C.13)

For the case of a dielectric medium,

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

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

$\displaystyle \frac{\partial E_x}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\frac{\partial P_x}{\partial t} + \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$ (C.15)
$\displaystyle \frac{\partial E_y}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\frac{\partial P_y}{\partial t} + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$ (C.16)
$\displaystyle \frac{\partial E_z}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\frac{\partial P_z}{\partial t} + \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$ (C.17)
$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$ (C.18)
$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$ (C.19)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right).$ (C.20)

Now, the electric displacement is defined

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

Moreover, in a linear dielectric medium,

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

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

$\displaystyle \frac{\partial D_x}{\partial t}$ $\displaystyle = \frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z},$ (C.23)
$\displaystyle \frac{\partial D_y}{\partial t}$ $\displaystyle = \frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x},$ (C.24)
$\displaystyle \frac{\partial D_z}{\partial t}$ $\displaystyle = \frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y},$ (C.25)
$\displaystyle \frac{\partial B_x}{\partial t}$ $\displaystyle = \frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y},$ (C.26)
$\displaystyle \frac{\partial B_y}{\partial t}$ $\displaystyle = \frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z},$ (C.27)
$\displaystyle \frac{\partial B_z}{\partial t}$ $\displaystyle = \frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x},$ (C.28)

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

$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle = v^{\,2}\left(\frac{\partial D_y}{\partial z}-\frac{\partial D_z}{\partial y}\right),$ (C.29)
$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle =v^{\,2}\left(\frac{\partial D_z}{\partial x}- \frac{\partial D_x}{\partial z}\right),$ (C.30)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle = v^{\,2}\left(\frac{\partial D_x}{\partial y}-\frac{\partial D_y}{\partial x}\right),$ (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,

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

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

$\displaystyle \frac{\partial E_x}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\sigma\,E_x+ \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$ (C.33)
$\displaystyle \frac{\partial E_y}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\sigma\,E_y + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$ (C.34)
$\displaystyle \frac{\partial E_z}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\sigma\,E_z+ \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$ (C.35)
$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$ (C.36)
$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$ (C.37)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right)$ (C.38)

The energy flux associated with an electromagnetic wave has the components

$\displaystyle {\cal I}_x$ $\displaystyle = E_y\,H_z-E_z\,H_y,$ (C.39)
$\displaystyle {\cal I}_y$ $\displaystyle = E_z\,H_x-E_x\,H_z,$ (C.40)
$\displaystyle {\cal I}_z$ $\displaystyle = E_x\,H_y-E_y\,H_x,$ (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

$\displaystyle [E_x]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (C.42)
$\displaystyle [E_y]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (C.43)
$\displaystyle [D_z]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (C.44)
$\displaystyle [H_x]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (C.45)
$\displaystyle [H_y]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (C.46)
$\displaystyle [H_z]_{z=0_-}^{z=0_+}$ $\displaystyle = 0$ (C.47)

(Fitzpatrick 2008).

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

$\displaystyle m\,\frac{d^{\,2}x}{dt^{\,2}}$ $\displaystyle = q\left(E_x + B_z\,\frac{d y}{d t}-B_y\,\frac{d z}{d t}\right),$ (C.48)
$\displaystyle m\,\frac{d^{\,2}y}{dt^{\,2}}$ $\displaystyle = q\left(E_y + B_x\,\frac{dz}{d t}-B_z\,\frac{d x}{d t}\right),$ (C.49)
$\displaystyle m\,\frac{d^{\,2}z}{dt^{\,2}}$ $\displaystyle = q\left(E_z + B_y\,\frac{d x}{d t}-B_x\,\frac{d y}{d t}\right),$ (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).

next up previous contents
Next: Bibliography Up: Oscillations and Waves Previous: Trigonometric Identities   Contents