The electric field in a certain region is given by the formula $\vec{E} = (\frac{K}{x^3}) \hat{i}$. What is the dimensional formula of $K$?

The work done by a gas molecule in an isolated system is given by $W = \alpha \beta^{2} e^{-\frac{x^{2}}{\alpha kT}}$,where $x$ is the displacement,$k$ is the Boltzmann constant,$T$ is the temperature,and $\alpha$ and $\beta$ are constants. Then the dimension of $\beta$ will be:

The equation of a circle is given by $x^2+y^2=a^2$,where $a$ is the radius. If the equation is modified to change the origin to a point other than $(0,0)$,find the correct dimensions of $A$ and $B$ in the new equation: $(x-At)^2+(y-\frac{t}{B})^2=a^2$. The dimensions of $t$ are given as $[T^{-1}]$.

Due to an explosion underneath water, a bubble started oscillating. If this oscillation has a time period $T$, which is proportional to $p^\alpha S^\beta E^\gamma$, where $p$ is static pressure, $S$ is the density of water, and $E$ is the total energy of the explosion, determine $\alpha, \beta$, and $\gamma$.

If $A$ and $B$ are two physical quantities having different dimensions,then which of the following cannot denote a physical quantity?

$A$ 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 $2$.