How do I find the dimensions of this quantity (in $cgs$)... $$\frac{4\pi me^2}{h^2n_o^{1/3}}$$
where $m$ is the mass of electron
$e$ is the magnitude of electronic charge
$h$ is the Planck's constant &
$n_o$ is equilibrium number density of electrons
There are various extensions of CGS to electromagnetism. Because $e^2$ appears in your expression without any speed of light etc., most likely, the electrostatic units – the most widespread ones – were used. In those units, the constant in the Coulomb's law known as $1/4\pi\epsilon$ in the SI units is set to one.
Equivalently, $e^2$ in our expression would get translated to $e^2/4\pi\epsilon$ in the SI units which is "electrostatic energy" except that $1/r$ is missing, so the units of $e^2$ are those of "energy times distance". However, there is one more factor of distance, $1/n_0^{1/3}$ (its inverted inverse volume, i.e. volume, to the power of 1/3, i.e. distance), so we have "energy times distance squared".
There is an additional factor of $m/\hbar^2$. Now, let's calculate the powers of centimeters, grams, and seconds. Grams cancel: one energy, one mass in the numerator; two Planck's constants in the denominator. The rest is "centimeter squared over (centimeter squared over second) squared" = "squared second over squared centimeter".
So the expression has the same units as $1/v^2$ where $v$ is a velocity. If some other CGS extension was used, it's plausible that the right units are another power of the velocity and the expression may even be dimensionless. The difference CGS extensions differ by the powers of the speed of light.
