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If I have let’s say an ionized gas, composed purely of singly charged positive ions; what would be the permitivity of the material? Is it infinite, as is the case with a perfect conductor, or am I missing out on something here?

Let’s say I place this ionized gas in a STATIC electric field, then the ions would rearrange in such a way to cancel the field, and hence the permitivity constant is essentially infinite - is my reasoning correct?

Now consider the case with an OSCILLATING electric field - how would the dielectric permitivity vary then for the ionized gas?

Just to be clear, I am NOT referring to a quasi-neutral plasma: I am referring to let’s say a cloud of O+ ions, with the ejected electrons completely removed from the cloud.

Joeseph123
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3 Answers3

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The equation of motion for ions is: $$ m \frac{\partial \overrightarrow{V} }{\partial t}=q~ \overrightarrow{E} $$ From which we get : $$ \overrightarrow{V}= \frac{q}{-i \omega t} \overrightarrow{E}$$

The ions current is then: $$ \overrightarrow{j}= n~q~ \overrightarrow{V}= \frac{n~q^{2}}{-i \omega t} \overrightarrow{E}= \sigma \overrightarrow{E}$$

From which, we get:$$ \varepsilon =1+ \frac{i~ \sigma }{ \omega \varepsilon _{0}}=1-\frac{n~q^{2}}{m~\varepsilon _{0}\omega ^{2}}=1- \frac{ \omega _{p}^{2} }{ \omega^{2} } $$

The permittivity depends on the frequency of the incoming wave and goes to infinity as $\omega$ goes to zero. However the model is very crude, if you include thermal effect, you get:

$$ \varepsilon = \begin{bmatrix}1- \frac{ \omega _{p}^{2} }{ \omega^{2} } & 0&0 \\0 & 1- \frac{ \omega _{p}^{2} }{ \omega^{2} }&0\\0&0&1- \frac{ \omega _{p}^{2} }{ \omega^{2}- \frac{k^{2}P_{0} \gamma }{m.n} } \end{bmatrix}$$ Even if $\omega =0$ there is still a non infinite term in the z direction(direction of the incoming wave).

Shaktyai
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I am referring to let’s say a cloud of O+ ions, with the ejected electrons completely removed from the cloud.

You're describing a bomb. This system will undergo what's known as a Coulomb explosion, and if the quantity of ions in this cloud is macroscopically significant, then the amount of energy that this explosion will detonate is on magnitudes that you only reach with nuclear weapons.

Worrying about the behaviour of this system under external fields is pretty much completely pointless.

Emilio Pisanty
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There are two cases for plasma (which would be your gas if it had electrons):

  1. low frequency EM waves, the permettivity is infinite meaning that these EM waves would reflect from the edge of this media

  2. high frequency EM waves, the permettivity is lower then infinite (like metals), so these EM waves could propagate through this gas

Note, from Eq. (1151), that the plasma frequency is proportional to the square root of the number density of free electrons.

So basically if this gas (which has no electrons) could exist for some time, and not explode, then EM waves could propagate in it (like in air), permettivity would be around 1.

The dialectric constant means the ability of a medium to separate its charges inside it. Since your gas has no separable charges, it has no ability to separate them, and does not need to separate them at all. EM waves could propagate in it like in air (close to vacuum).

http://farside.ph.utexas.edu/teaching/em/lectures/node100.html

Árpád Szendrei
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