Dragon Notes

UNDER CONSTRUCTION
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Normal Incidence [PEC]

When normally incident on a Perfect Electrical Conductor boundary, the incident EM-wave described in phasor form by

${\overrightarrow{\boldsymbol{E}_r}}\left(z\right)=\widehat{\boldsymbol{x}}E_{i0}e^{-j{\beta }_1z},\ \ \ {\overrightarrow{\boldsymbol{H}_i}}\left(z\right)=\widehat{\boldsymbol{y}}\frac{E_{i0}}{{\eta }_1}e^{-j{\beta }_1z}_1$

is totally reflected off the boundary as a wave described by

${\overrightarrow{\boldsymbol{E}_r}}\left(z\right)=\widehat{\boldsymbol{x}}E_{i0}e^{+j{\beta }_1z}_1,\ {\overrightarrow{\boldsymbol{H}_r}}\left(z\right)=\widehat{\boldsymbol{y}}\frac{E_{i0}}{{\eta }_1}e^{+j{\beta }_1z}.$

The above describe waves with equal wave numbers and frequencies (not shown) propagating in opposite directions. These incident and reflected waves sum to the total EM-wave in medium 1 – a standing wave:

${\overrightarrow{\boldsymbol{E}_1}}\left(z,t\right)=\widehat{\boldsymbol{x}}2E_{i0}{\mathrm{sin} \left({\beta }_1z\right)\ }{\mathrm{sin} \left(\omega t\right)\ },\ \ \ {\overrightarrow{\boldsymbol{H}_1}}\left(z,t\right)=\widehat{\boldsymbol{y}}2\frac{E_{i0}}{{\eta }_1}{\mathrm{cos} \left({\beta }_1z\right)\ }{\mathrm{cos} \left(\omega t\right)\ }.$

The comprising $$E$$- and $$H$$- fields have zeros and maxima at fixed distances from the boundary for all times $$t$$:

$\left. \begin{array}{c} \mathrm{Zeros\ of\ }{\overrightarrow{\boldsymbol{E}_1}}\left(z,t\right) \\ \mathrm{Maxima\ of\ }{\overrightarrow{\boldsymbol{H}_1}}\left(z,t\right) \end{array} \right\}\mathrm{occur\ at}\ \boxed{z=-n\frac{\lambda }{2},\ n=0,1,2,\dots}$ $\left. \begin{array}{c} \mathrm{Maxima\ of\ }{\overrightarrow{\boldsymbol{E}_1}}\left(z,t\right) \\ \mathrm{Zeros\ of\ }{\overrightarrow{\boldsymbol{H}_1}}\left(z,t\right) \end{array} \right\}\mathrm{occur\ at}\ \boxed{z=-\left(2n+1\right)\frac{\lambda }{4},\ n=0,1,2,\dots}$

The incident $$H$$- field will induce a current on the boundary surface – equal to the amount of discontinuity in the field (as it vanishes inside the PEC):

${\overrightarrow{\boldsymbol{J}_s}}\left(z,t\right)=\widehat{\boldsymbol{n}}_2\times {\overrightarrow{\boldsymbol{H}_1}}\left(0,t\right)$