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Electromagnetic Theory

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

$\displaystyle \frac{\partial E_x}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(j_x + \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$ (1294)
$\displaystyle \frac{\partial E_y}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(j_y + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$ (1295)
$\displaystyle \frac{\partial E_z}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(j_z + \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$ (1296)
$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$ (1297)
$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$ (1298)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right)$ (1299)

(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,

$\displaystyle {\bf j} = {\bf0}.$ (1300)

Hence, the previous equations simplify to give

$\displaystyle \frac{\partial E_x}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$ (1301)
$\displaystyle \frac{\partial E_y}{\partial t}$ $\displaystyle = -\frac{1}{\epsilon_0}\left(\frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$ (1302)
$\displaystyle \frac{\partial E_z}{\partial t}$ $\displaystyle = -\frac{1}{\epsilon_0}\left(\frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$ (1303)
$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$ (1304)
$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$ (1305)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right).$ (1306)

For the case of a dielectric medium,

$\displaystyle {\bf j} = \frac{\partial{\bf P}}{\partial t},$ (1307)

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

$\displaystyle \frac{\partial E_x}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\frac{\partial P_x}{\partial t} + \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$ (1308)
$\displaystyle \frac{\partial E_y}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\frac{\partial P_y}{\partial t} + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$ (1309)
$\displaystyle \frac{\partial E_z}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\frac{\partial P_z}{\partial t} + \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$ (1310)
$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$ (1311)
$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$ (1312)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right).$ (1313)

In a linear dielectric medium,

$\displaystyle {\bf P} = \epsilon_0\,(\epsilon-1)\,{\bf E},$ (1314)

where $ \epsilon$ is the relative dielectric constant (Fitzpatrick 2008). In this case, defining the electric displacement,

$\displaystyle {\bf D} = \epsilon_0\,\epsilon\,{\bf E},$ (1315)

we get

$\displaystyle \frac{\partial D_x}{\partial t}$ $\displaystyle =-\left(\frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$ (1316)
$\displaystyle \frac{\partial D_y}{\partial t}$ $\displaystyle =-\left( \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$ (1317)
$\displaystyle \frac{\partial D_z}{\partial t}$ $\displaystyle =-\left(\frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$ (1318)
$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle =v^2\left(\frac{\partial D_y}{\partial z}-\frac{\partial D_z}{\partial y}\right),$ (1319)
$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle =v^2\left(\frac{\partial D_z}{\partial x}- \frac{\partial D_x}{\partial z}\right),$ (1320)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle =v^2\left(\frac{\partial D_x}{\partial y}-\frac{\partial D_y}{\partial x}\right),$ (1321)

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

In an Ohmic conductor,

$\displaystyle {\bf j} = \sigma {\bf E},$ (1322)

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

$\displaystyle \frac{\partial E_x}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\sigma E_x+ \frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$ (1323)
$\displaystyle \frac{\partial E_y}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\sigma E_y + \frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$ (1324)
$\displaystyle \frac{\partial E_z}{\partial t}$ $\displaystyle =-\frac{1}{\epsilon_0}\left(\sigma E_z+ \frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$ (1325)
$\displaystyle \frac{\partial H_x}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$ (1326)
$\displaystyle \frac{\partial H_y}{\partial t}$ $\displaystyle = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$ (1327)
$\displaystyle \frac{\partial H_z}{\partial t}$ $\displaystyle =\frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right)$ (1328)

The energy flux associated with an electromagnetic wave has the components

$\displaystyle {\cal I}_x$ $\displaystyle = E_y\,H_z-E_z\,H_y,$ (1329)
$\displaystyle {\cal I}_y$ $\displaystyle = E_z\,H_x-E_x\,H_z,$ (1330)
$\displaystyle {\cal I}_z$ $\displaystyle = E_x\,H_y-E_y\,H_x,$ (1331)

irrespective of the medium (as long as it is non-magnetic) (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

$\displaystyle [E_x]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (1332)
$\displaystyle [E_y]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (1333)
$\displaystyle [D_z]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (1334)
$\displaystyle [H_x]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (1335)
$\displaystyle [H_y]_{z=0_-}^{z=0_+}$ $\displaystyle =0,$ (1336)
$\displaystyle [H_z]_{z=0_-}^{z=0_+}$ $\displaystyle = 0$ (1337)

(Fitzpatrick 2008).

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

$\displaystyle m\,\frac{d^2x}{dt^2}$ $\displaystyle = q\left(E_x + B_z \frac{d y}{d t}-B_y \frac{d z}{d t}\right),$ (1338)
$\displaystyle m\,\frac{d^2y}{dt^2}$ $\displaystyle = q\left(E_y + B_x \frac{dz}{d t}-B_z \frac{d x}{d t}\right),$ (1339)
$\displaystyle m\,\frac{d^2z}{dt^2}$ $\displaystyle = q\left(E_z + B_y \frac{d x}{d t}-B_x \frac{d y}{d t}\right),$ (1340)

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
Next: Bibliography Up: Oscillations and Waves Previous: Trigonometric Identities
Richard Fitzpatrick 2013-04-08