11.1: Introduction
- Page ID
- 22727
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 Ex = 0 and Hy = 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).
