5.1 The Wave Equation

Note : This section involves some vector calculus, which is really beyond the scope of this course. I include this only to give you a flavor for the theory. Once we have the wave equation, we will return to familiar territory.

Maxwell's equations describing electric and magnetic fields in vacuum expressed in differential form are

$\begin{displaymath} \begin{array}{lcl} \vec{\nabla} \cdot \vec{E} = \frac{\rho}{... ...mu_o \epsilon_o \frac{\partial \vec{E}}{\partial t} \end{array}\end{displaymath}$

(5.1)

where $\epsilon_o$ is the permittivity of free space, $\mu_o$ is the permeability of free space, $\rho$ is the charge density, and $\vec{J}$ is the current density.

We will limit our discussion of electromagnetic waves to regions of space free of charges and currents, for which Maxwell's equations take the form,

$\begin{displaymath} \begin{array}{lcl} \vec{\nabla} \cdot \vec{E} = 0 && \vec{\n... ...mu_o \epsilon_o \frac{\partial \vec{E}}{\partial t} \end{array}\end{displaymath}$

(5.2)

Taking the curl of the cross product equations yields

$\displaystyle \vec{\nabla} \times (\vec{\nabla} \times \vec{E})$	$\textstyle =$	$\displaystyle -\frac{\partial}{\partial t}(\vec{\nabla} \times \vec{B})$
$\displaystyle \vec{\nabla} \times (\vec{\nabla} \times \vec{B})$	$\textstyle =$	$\displaystyle \mu_o \epsilon_o \frac{\partial}{\partial t} (\vec{\nabla} \times \vec{B})$	(5.3)

With Eq.'s 5.2 and the vector calculus identity

$\begin{displaymath} \vec{\nabla} \times (\vec{\nabla} \times \vec{A}) = \vec{\nabla}(\vec{\nabla} \cdot \vec{A}) - \nabla^2 \vec{A} \end{displaymath}$

(5.4)

where $\nabla^2$ is shorthand for $\vec{\nabla} \cdot \vec{\nabla}$ , Eq.'s 5.3 become

$\displaystyle \nabla^2 \vec{E}$	$\textstyle =$	$\displaystyle \mu_o \epsilon_o \frac{\partial^2 \vec{E}}{\partial t^2}$
$\displaystyle \nabla^2 \vec{B}$	$\textstyle =$	$\displaystyle \mu_o \epsilon_o \frac{\partial^2 \vec{B}}{\partial t^2}$	(5.5)

With three spatial components for each field, we have six separate wave equations.

However for plane waves propagating along $\hat{\imath}$ , several of these wave equations vanish, and we have

$\displaystyle \frac{\partial^2 \vec{E}}{\partial x^2}$	$\textstyle =$	$\displaystyle \mu_o \epsilon_o \frac{\partial^2 \vec{E}}{\partial t^2}$
$\displaystyle \frac{\partial^2 \vec{B}}{\partial x^2}$	$\textstyle =$	$\displaystyle \mu_o \epsilon_o \frac{\partial^2 \vec{B}}{\partial t^2}$	(5.6)

The electric and magnetic fields in free space obey the classical wave equation exactly. This is the first system for which we have not made some kind of small amplitude approximation.

Comparing Eq. 5.6 with Eq. 1.69, we identify the phase velocity

$\begin{displaymath} v_p = \frac{1}{\sqrt{\epsilon_o \mu_o}} \end{displaymath}$

(5.7)

of light in vacuum $c = v_p = 3.00\mathrm{e}{8}$ m/s.