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Many books on plasma physics (Chen, Goldston, Lieberman) say that quasineutrality must be satisfied for the matter in question to be a plasma. Yet, we know that non-neutral plasmas exist. So why do so many books make this claim?

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In the old days, most plasmas were created with discharges in gases or such. This can stay a plasma for a long time (at least seconds). If it were not quasi-neutral, the accumulated charge would strongly pull ions away from electrons and make the plasmas explode. That's why books make this claim, and it is still true in many cases.

Nowadays, plasmas can be created in ways that the explosion time is larger than the time it takes to create it. For example, you can create plasmas in less than a picosecond with lasers, and these plasmas may be non-neutral.

Another way is to confine a non-neutral plasma with electric and magnetic fields.

Note that some people like to call plasmas only neutral ones, or even call them plasmas only when it is fully ionized. It is just a matter of definition, and old books can be confusing.

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I don't think that quasineutrality is "a required condition", but is a property of most plasmas.

Quasineutrality is the tendency for plasmas to attain electrical neutrality. It means that if a neutral plasma (with equal numbers of electrons and ions), should generate a region with a slight electrical charge, then because (a) the electric force is huge ($10^{39}$ time greater than the gravitational force), and (b) the charges are very mobile, then charges would tend to move to cancel out any charge imbalance.

The description suggests that this is a "tendency" because quasineutrality can be violated, and charged regions can be found in "double layers" and "particle beams".

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    $\begingroup$ Dear iantresman. It is usually frown upon to directly copy-paste identical answers. (The problem is if everybody start to copy-paste identical answers en mass.) In general in such situations, please consider one of the following options: (i) Delete one of your answers. (ii) Flag for duplicate posts and delete one of your answers. (iii) If you think the two posts are not duplicates, then personalize each answer to address the two different specific questions. $\endgroup$
    – Qmechanic
    Dec 4, 2014 at 23:14
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Quasi-neutrality is only observed with bulk plasma. Although it appears neutral to outside observance at large scales, we also know at smaller scales the movement of the charge carriers creates EM fields and electric currents which further affect its behavior. In reality in space no such condition exists, as the plasma is not confined in little glass jars and is in constant motion.

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