Normal incidence on a perfect conductor

whenever an EM Wave travelling in one medium impinges second medium the wave gets partially transmitted and partially reflected depending up on the type of the second medium.

Assume the first case in Normal incidence that is Normal incidence on a Perfect conductor.

i.e an EM wave propagating in free space strikes suddenly a conducting Boundary which means the other medium is a conductor.

The figure shows a plane Wave which is incident normally upon a boundary between free space and a perfect conductor.

assume the wave is propagating in positive z-axis and the boundary is z=0 plane.

The transmitted wave E_{t}=0 since the electric field intensity inside a perfect conductor is zero.

The incident  E_{i}(t) and reflected  E_{r}(t) waves are in the medium 1  that is free space.

The energy transmitted is zero so the energy absorbed by the conductor is zero and entire wave is reflected to the same medium

now incident wave is E_{i}=E_{i}e^{-\gamma z}

\because \alpha =0  in free space \beta =\beta _{1} for medium 1

\overrightarrow{E_{i}} = E_{i}e^{-j\beta z} (-\beta _{1}z a wave propagating in positive z-direction) and the reflected wave is \overrightarrow{E_{r}} = E_{r}e^{j\beta z} (\beta _{1}z a wave propagating in positive z-direction).

\overrightarrow{E}_{total} = E_{i}e^{-j\beta z}+ E_{r}e^{j\beta z} .

by using tangential components {E}_{tan1} = E_{tan2} .

{E}_{i} +{E}_{r} = 0

{E}_{i} =-{E}_{r}

The resultant wave is  \overrightarrow{E}_{total} = E_{i}e^{-j\beta_{1} z}+ E_{r}e^{j\beta_{1} z} .

\overrightarrow{E}_{total} = E_{i}e^{-j\beta_{1} z}- E_{i}e^{j\beta_{1} z} .

\overrightarrow{E}_{total}(z) = -2E_{i}\ j \sin \beta_{1} z

the above equation is in phasor notation , converting the above equation into time-harmonic (or) sinusoidal variations

\widetilde{{E}_{total}}(z,t) = Re\left \{ -2E_{i}\ j \sin \beta_{1} z \ e^{j\omega t }\right \}

\widetilde{{E}_{total}}(z,t) = 2E_{i} \sin \beta_{1} z \ \sin \omega t .

This is the wave equation which represents standing wave , which is the contribution of incident and reflected waves. as this wave is stationary it does not progress.

it has maximum amplitude at odd multiples of  \frac{\lambda }{4} and minimum amplitude at multiples of \frac{\lambda }{2} .

Similarly The resultant Magnetic field is

The resultant wave is  \overrightarrow{H}_{total} = H_{i}e^{-j\beta_{1} z}+ H_{r}e^{j\beta_{1} z} .

\overrightarrow{H}_{total} = H_{i}e^{-j\beta_{1} z}+ H_{i}e^{j\beta_{1} z} .

\overrightarrow{H}_{total}(z) = 2H_{i}\ \cos \beta_{1} z

the above equation is in phasor notation , converting the above equation into time-harmonic (or) sinusoidal variations

\widetilde{{H}_{total}}(z,t) = Re\left \{ 2H_{i}\ \cos \beta_{1} z \ e^{j\omega t }\right \}

\widetilde{{H}_{total}}(z,t) = 2H_{i} \cos \beta_{1} z \ \cos \omega t .

this wave is  a stationary wave  it has minimum amplitude at odd multiples of  \frac{\lambda }{4} and maximum amplitude at multiples of \frac{\lambda }{2} .

 

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Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.