5.2 Plane Waves

A plane wave solution to Eq. 5.6 traveling in $\hat{\imath}$ takes the form^5.1

$\displaystyle \vec{E}$	$\textstyle =$	$\displaystyle \vec{E}_o e^{i(kx - \omega t + \phi)}$
$\displaystyle \vec{B}$	$\textstyle =$	$\displaystyle \vec{B}_o e^{i(kx - \omega t + \phi)}$	(5.8)

and Maxwell's equations place constraints on the directions of $\vec{E}_o$ and $\vec{B}_o$ relative to each other and to $\hat{\imath}$ , the direction of propagation.

$\vec{\nabla} \cdot \vec{E} = \vec{\nabla} \cdot \vec{B} = 0$ are here $\frac{\partial E_x}{\partial x} = \frac{\partial B_x}{\partial x} = 0$ and require that

and are constant. (In the absence of a static field, they are zero.)
$\vec{E}_o$ and $\vec{B}_o$ are perpendicular to $\hat{\imath}$ , the direction of propagation.

$\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ gives

$\displaystyle \vec{\nabla} \times \vec{E}$	$\textstyle =$	$\displaystyle \left( \frac{\partial}{\partial x} \hat{\imath} +\frac{\partial}{... ...\times (E_{oy} \hat{\jmath} + E_{oz} \hat{k}) e^{i(kx - \omega t + \phi)}$
	$\textstyle =$	$\displaystyle (-\frac{\partial}{\partial x} E_{oz} \hat{\jmath} +\frac{\partial}{\partial x} E_{oy} \hat{k} ) e^{i(kx - \omega t + \phi)}$
	$\textstyle =$	$\displaystyle ik(-E_{oz} \hat{\jmath} + E_{oy} \hat{k} ) e^{i(kx - \omega t + \phi)}$
	$\textstyle =$	$\displaystyle i\omega (B_{oy} \hat{\jmath} + B_{oz} \hat{k}) e^{i(kx - \omega t + \phi)}$	(5.9)

which with $\vec{k} = k \hat{\imath}$ we recognize as

$\begin{displaymath} \vec{B}_o = \frac{\vec{k} \times \vec{E}_o}{\omega} \end{displaymath}$

(5.10)

We found above that $\vec{E}$ and $\vec{B}$ are perpendicular to $\hat{i}$ , which together with Eq. 5.10 relates the amplitudes of the fields,

$\begin{displaymath} \frac{E_o}{B_o} = \frac{\omega}{k} = c \end{displaymath}$

(5.11)

**Figure 5.1:** An electromagnetic plane wave. $\vec{E}$ and $\vec{B}$ are in phase, perpendicular to each other and to the wavenumber vector $\vec{k}$ along the direction of propagation.
$\scalebox{0.7}{ \includegraphics{em.eps} }$

In general,

$\displaystyle \vec{E}$	$\textstyle =$	$\displaystyle \vec{E}_o e^{i(\vec{k}\cdot\vec{r} - \omega t + \phi)}$
$\displaystyle \vec{B}$	$\textstyle =$	$\displaystyle \vec{B}_o e^{i(\vec{k}\cdot\vec{r} - \omega t + \phi)} = \frac{\vec{k} \times \vec{E}}{\omega}$	(5.12)

and $\vec{E}_o$ and $\vec{B}_o$ are perpendicular to each other and perpendicular to $\vec{k}$ as shown in Fig. 5.1.

It is possible to make $\vec{E}$ , $\vec{B}$ , and $\vec{k}$ mutually perpendicular and violate Eq. 5.10. Use the right hand rule to verify that Fig. 5.1 shows the correct relative orientations.