11.1: Introduction

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Consider a plane wave propagating along the z-direction in vacuum, and polarized with its electric vector along the x-axis: its magnetic field vector must be directed along the y-axis. Now introduce two infinitely conducting metal planes which block off all of space except the region between x= +a and x= -a, see Figure (11.1.1). The boundary conditions at x= ±a that must be satisfied by the electric and magnetic fields are

the tangential components of \(E\) must be zero;
the normal component of \(H\) must be zero.

This latter condition is a consequence of the Maxwell equation

\[\operatorname{div}(\overrightarrow{\mathrm{B}})=0 \nonumber\]

which requires the normal component of \(\vec B\) to be continuous through an interface, coupled with the requirement that both the electric and magnetic fields are zero inside a perfect conductor: recall from Chapter(10) that in the limit of infinite conductivity the skin depth of a metal goes to zero. Notice that the above two boundary conditions are satisfied by the plane wave. The plane wave solutions of Maxwell’s equations

\[\begin{align}
&\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{0} \exp (i[\mathrm{kz}-\omega \mathrm{t}]),\\
&\mathrm{H}_{\mathrm{y}}=\mathrm{H}_{0} \exp (i[\mathrm{kz}-\omega \mathrm{t}]),
\end{align}\]

can be used to describe the propagation of electromagnetic energy between two conducting planes. Energy is transported at the speed of light just as it is for a plane wave in free space. Notice that if it is attempted to close in the radiation with conducting planes at y= ±b the boundary conditions E_x = 0 and H_y = 0 cannot be satisfied on the planes y= ±b. Waves can be transmitted through such hollow pipes but the radiation bounces from wall to wall in a complex pattern that will be studied later. It will be shown that waves cannot be transmitted through a hollow pipe if the frequency is too low; there exists a lower frequency cut-off. However, a pair of parallel conducting planes unbounded in one transverse direction can transmit waves at all frequencies. In practice infinite planes are inconvenient, so one uses either strip-lines or co-axial cables, see Figures (11.1.2) and (11.2.3).

Figure 11.1.PNG — Figure \(\PageIndex{1}\): A plane wave propagating between two perfectly conducting planes. \(\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{0} \exp (i[\mathrm{kz}-\omega \mathrm{t}]), \mathrm{H}_{\mathrm{y}}=\left(\mathrm{E}_{0} / \mathrm{Z}_{0}\right) \exp (i[\mathrm{kz}-\omega \mathrm{t}])\).

Figure 11.2.PNG — Figure \(\PageIndex{2}\): A strip-line. \(\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{0}, \mathrm{V}=\mathrm{E}_{0} \mathrm{d} . \mathrm{H}_{\mathrm{y}}=\mathrm{E}_{0} / \mathrm{Z}_{0}, \mathrm{I}=\mathrm{w} \mathrm{J}_{\mathrm{s}}=\mathrm{w}\left(\mathrm{E}_{0} / \mathrm{Z}_{0}\right)\)