Electromagnetic wave in vacuum

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The electromagnetic wave in a vacuum is studied :

\(\vec E = E_0 \;cos(\alpha z)sin(\omega t - kx)\;\vec u_y\)

Question

Is the corresponding wave plane ? Progressive ? Harmonic ? To justify.

What does it remind you of ?

Solution

It is not plane, but progressive and harmonic.

It is a wave that may be encountered in .

Question

Calculate the magnetic field.

Solution

We can't use the relationship of structure because the wave is not a plane wave.

We calculate the magnetic field from the Maxwell-Faraday equation :

\(\overrightarrow{\mathrm{rot}} \vec E = -\frac{\partial \vec B}{\partial t}\)

We find :

\(\begin{array}{l}{B_x} = \frac{{{E_0}\alpha }}{\omega }\sin \alpha z\cos (\omega t - kx) \\{B_z} = \frac{{k{E_0}}}{\omega }\cos \alpha z\sin (\omega t - kx) \\\end{array}\)

Question

Is there any dispersion ?

Solution

Determining the dispersion relation.

For this, we use the wave equation of the electric field in vacuum (d'Alembert's equation) :

\(\Delta \vec E - \frac {1}{c^2}\frac{{{\partial ^2}\vec E}}{{\partial {t^2}}} = \vec 0\)

We find :

\({k^2} = \frac{{{\omega ^2}}}{{{c^2}}} - {\alpha ^2}\)

There is dispersion with a phase velocity that is : (we assume \(\omega/c > \alpha\))

\({v_\varphi } = \frac{c}{{\sqrt {1 - \frac{{{\alpha ^2}{c^2}}}{{{\omega ^2}}}} }}\)