The relation between the maximum electron density ${{N}_{\max }}$ and the critical frequency ${{f}_{c}}$ for the ionosphere can be given as:
$A.{{f}_{c}}=\sqrt{9{{N}_{\max }}}$
$B.{{f}_{c}}=\sqrt{9}{{N}_{\max }}$
$C.{{f}_{c.}}=9\sqrt{{{N}_{\max }}}$
$D.$ None of these.

Hint: We have to use the idea of critical frequency for the ionosphere. Critical frequency is defined as the frequency above which the waves penetrate the ionosphere and below which the waves are reflected back from the ionosphere. Electron density is defined as the measure of the probability of the presence of an electron at an infinitesimal element of space surrounding any given point.

Complete step-by-step solution
We should know that critical frequency is defined as the frequency above which the waves penetrate the ionosphere and below which the waves are reflected back from the ionosphere. Critical frequency is also called limiting frequency.
We know that the maximum electron density of the ionosphere is given by the following relation:-
${{f}_{c.}}=9\sqrt{{{N}_{\max }}}$ . Which can also be written as ${{f}_{c}}=9N_{\max }^{\dfrac{1}{2}}$. Where ${{f}_{c}}$ denotes critical frequency and ${{N}_{\max }}$ denotes electron density per ${{m}^{3}}$ . The Ionosphere transmits a range of frequencies between \[0.1\] to $30MHz$ .
Therefore, from the above given options only option $(C)$is correct.
Additional Information:
The ionized part of earth’s upper atmosphere is called the ionosphere. This region includes the thermosphere, parts of the exosphere and mesosphere. The source for ionization for the ionosphere is solar radiation. Radio propagation is influenced by the ionosphere to far places on earth and therefore it has very high importance. It is from $48km$ to $965km$ in the atmosphere. We have following diagram to show the different layers of the atmosphere:-


Note: We should remember the fact that the ionosphere is not a separate layer of atmosphere but the space in the thermosphere, parts of the exosphere, and mesosphere which is ionized in nature. It should be noted that below critical frequency radio waves are reflected by the ionosphere. The Ionosphere is composed of electrons and electrically charged atoms and molecules that surround the earth.