Reflection on a metal
Take 10 minutes to prepare this exercise.
Then, if you lack ideas to begin, look at the given clue and start searching for the solution.
A detailed solution is then proposed to you.
If you have more questions, feel free to ask them on the forum.
A MPPW is normally sent to a perfectly conductor plane, located in the half-space \(z>0\). The space \(z<0\) being vacuum.
The polarization of this wave is circular and the associated electric field can be written in real notation :
\(E_{x_i}=E_0 cos(\omega t - k_iz+\pi /2)\)
\(E_{y_i}=E_0 cos(\omega t - k_iz)\)
Question
Determining the resultant wave structure in vacuum (for \(z<0\)).
Solution
The total electric field must be zero at the surface (for \(z=0\)), therefore the reflected electric field is :
\({E_{x,r}} = - {E_0}\cos (\omega t + {k_i}z + \frac{\pi }{2})\;\;\;\;;\;\;\;\;{E_{y,r}} = - {E_0}\cos (\omega t + {k_i}z)\)
The structure of the total electric field above metal is :
\(\begin{array}{l}{E_x} = {E_{x,i}} + {E_{x,r}} = {E_0}\left[ {\cos (\omega t - {k_i}z + \frac{\pi }{2}) - \cos (\omega t + {k_i}z + \frac{\pi }{2})} \right] = 2{E_0}\cos \omega t\sin {k_i}z \\{E_y} = {E_{y,i}} + {E_{y,r}} = {E_0}\left[ {\cos (\omega t - {k_i}z) - \cos (\omega t + {k_i}z)} \right] = 2{E_0}\sin \omega t\sin {k_i}z \\\end{array}\)
A standing wave is obtained.
The magnetic field is obtained with Maxwell-Faraday equation :
\(\overrightarrow {rot} \vec E = \left| \begin{array}{l}0 \\0 \\\partial /\partial z \\\end{array} \right. \wedge \left| \begin{array}{l}2{E_0}\cos \omega t\sin {k_i}z \\2{E_0}\sin \omega t\sin {k_i}z \\0 \\\end{array} \right. = \left| \begin{array}{l}- 2{E_0}{k_i}\sin \omega t\cos {k_i}z \\ 2{E_0}{k_i}\cos \omega t\cos {k_i}z \\0 \\\end{array} \right. = - \frac{{\partial \vec B}}{{\partial t}}\)
Where :
\(\vec B = \left| \begin{array}{l}- 2\frac{{{E_0}{k_i}}}{\omega }\cos \omega t\cos {k_i}z = - 2\frac{{{E_0}}}{c}\cos \omega t\cos {k_i}z \\-2\frac{{{E_0}{k_i}}}{\omega }\sin \omega t\cos {k_i}z = -2\frac{{{E_0}}}{c}\sin \omega t\cos {k_i}z \\0 \\\end{array} \right.\)
We can calculate Poynting vector and verify that its average value is zero : a standing wave does not transport energy.