In the relation $y = a \cos (\omega t - kx)$,the dimensional formula for $k$ is

$A$ dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let $N$ be the number density of free electrons, each of mass $m$. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency $\omega_p$, which is called the plasma frequency. To sustain the oscillations, a time-varying electric field needs to be applied that has an angular frequency $\omega$, where a part of the energy is absorbed and a part of it is reflected. As $\omega$ approaches $\omega_p$, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
$1.$ Taking the electronic charge as $e$ and the permittivity as $\varepsilon_0$, use dimensional analysis to determine the correct expression for $\omega_p$.
$(A) \sqrt{\frac{N e}{m \varepsilon_0}}$ $(B) \sqrt{\frac{m \varepsilon_0}{N e}}$ $(C) \sqrt{\frac{N e^2}{m \varepsilon_0}}$ $(D) \sqrt{\frac{m \varepsilon_0}{N e^2}}$
$2.$ Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $N \approx 4 \times 10^{27} \ m^{-3}$. Take $\varepsilon_0 \approx 10^{-11}$ and $m \approx 10^{-30}$, where these quantities are in proper $SI$ units.
$(A) 800 \ nm$ $(B) 600 \ nm$ $(C) 300 \ nm$ $(D) 200 \ nm$
Give the answer for question $1$ and $2$.

Why does the concept of dimension have basic importance?

In a particular system of units,a physical quantity can be expressed in terms of the electric charge $e$,electron mass $m_e$,Planck's constant $h$,and Coulomb's constant $k = \frac{1}{4 \pi \epsilon_0}$,where $\epsilon_0$ is the permittivity of vacuum. In terms of these physical constants,the dimension of the magnetic field is $[B] = [e]^\alpha [m_e]^\beta [h]^\gamma [k]^\delta$. The value of $\alpha + \beta + \gamma + \delta$ is . . . . .

The speed of a water wave $v$ depends on its wavelength $\lambda$,the density of water $\rho$,and the acceleration due to gravity $g$. Determine the relationship between these quantities.

Assertion $(A)$: Energy per unit volume and angular momentum can be added dimensionally.
Reason $(R)$: Physical quantities having same dimensions can be added or subtracted.