Basically, my book (Modern Optics by Robert Guenther) presents the "proof" of the interdependancy of the plane waves E and B as follow:(adsbygoogle = window.adsbygoogle || []).push({});

Suppose [itex]\vec{E}[/itex] is an electric plane wave:

[tex]\vec{E} = \vec{E_0}e^{i(\omega t - \vec{k}\cdot \vec{r}+ \phi)}[/tex].

Then we find that

[tex]\frac{\partial \vec{E}}{\partial t}=i\omega \vec{E}[/tex].

And if [itex]\vec{B}[/itex] is a plane wave in-phase with [itex]\vec{E}[/itex], such as

[tex]\vec{B} = \vec{B_0}e^{i(\omega t - \vec{k}\cdot \vec{r}+ \phi)}[/tex],

then

[tex]\vec{\nabla}\times \vec{B} = -i\vec{k}\times \vec{B}[/tex].

And thus, given [itex]\vec{E}[/itex] a plane wave, [itex]\vec{B}[/itex] a plane in-phase satify the Maxwell equation

[tex]\vec{\nabla}\times \vec{B} = \mu\epsilon \frac{\partial \vec{E}}{\partial t}[/tex]

under the simple condition that [itex]E_0 = cB_0[/itex] but what tells me that given [itex]\vec{E}[/itex] a plane wave, this the only solution? It's this little detail that bugs me.

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# E-M waves detail

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