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According to "Introduction to Plasma Physics and Controlled Fusion 3rd Edition", in discussing electromagnetic waves in a plasma, $\vec{E}$ is said to be a transverse wave. Is $\vec{E}$ a transverse wave even in a plasma? I only know the proof that $\vec{E}$ is a transverse wave "in a vacuum". By one of Maxwell equations and by assuming a plane wave solution, we easily get

$\frac{\rho}{\epsilon _0} = \vec{\nabla} \cdot \vec{E} = \vec{k} \cdot \vec{E}.$

$\vec{E}$ is not a transverse wave, is it?

(In the book, an equation $\vec{k} \cdot \vec{E} = 0$ is used. So I have to know whether this equation is truly true.)

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    $\begingroup$ Given you are reading a book on plasma physics, have you tried just reading a little further? I suspect your question will be answered fairly soon. $\endgroup$ Nov 30 '17 at 13:08
  • $\begingroup$ @BySymmetry Thank you for your comment. I'm reviewing today's lesson, but the reason wasn't explained in the class and isn't found in the book. In my opinion, a plasma is always quasineutral so that $\rho$ is small and can be ignored. But I'm not confident. $\endgroup$
    – ynn
    Nov 30 '17 at 13:17
  • $\begingroup$ @ynn Your assessment is incorrect. $\rho=0$ is typically true in equilibrium, but localized perturbations in the charge density are characteristic of many wave modes in plasmas (though not all). $\endgroup$
    – J. Murray
    Nov 30 '17 at 13:19
  • $\begingroup$ @J.Murray Thank you. By "many wave modes", do you mean in a plasma $\vec{E}$ doesn't have to be a transverse wave? If so, why $\vec{k} \cdot \vec{E} = 0$ is used in the book? $\endgroup$
    – ynn
    Nov 30 '17 at 13:24
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    $\begingroup$ Plasmas support many, many different types of wave. By setting $\vec k \cdot \vec E=0$, you are restricting your attention to exclusively transverse waves. In an unmagnetized plasma, this is characteristic of electromagnetic waves (though electrostatic waves exist). $\endgroup$
    – J. Murray
    Nov 30 '17 at 13:35
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$\vec{E}$ is not a transverse wave, is it?

There is nothing wrong with transverse ($\mathbf{k} \cdot \mathbf{E} = 0$, e.g., O-mode) or longitudinal ($\mathbf{k} \times \mathbf{E} = 0$, e.g., ion acoustic waves) waves in a plasma.

Is $\vec{E}$ a transverse wave even in a plasma?

It can be, but again it is not required to be. There are lots of waves with longitudinal oscillations of $\mathbf{E}$ that exist in a plasma and lots of work done measuring the electric fields associated with these modes (e.g., the following spacecraft measure electric and magnetic fields Wind, THEMIS, MMS, etc.).

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