A student determines a dimensionless quantity, $\mathrm{B}= \frac{{\mathrm{e}}^{\mathrm{n}}}{2{\mathrm{\epsilon }}_0\mathrm{hc}}$. Findd the value of n. (here, e = electric charge, ${\mathrm{\epsilon }}_0$ electric permittivity of vacuum, h = Planck's constant and c = speed of light.) 

Dimension of B = $\left[{\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^0\right]$

Dimension e = [AT]

Dimensions ${\mathrm{\epsilon }}_0$= $\left[{\mathrm{A}}^2{\mathrm{M}}^{-1}{\mathrm{L}}^{-3}{\mathrm{T}}^4\right]$

Dimensions h = $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-1}\right]$

Dimension c = $\left[{\mathrm{LT}}^{-1}\right]$

$\therefore \left[M^0{L}^0{T}^0\right] = \frac{{\left[AT\right]}^n}{\left[A^2{M}^{-1}{L}^{-3}{T}^{-4}\right]\left[M{L}^2{T}^{-1}\right]\left[L{T}^{-1}\right]}$

or $\left[{\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^0\right] = \left[{\mathrm{A}}^{\mathrm{n}-2}{\mathrm{L}}^0{\mathrm{T}}^{\mathrm{n}-2}\right]$

$\therefore$n -2 = 0

$\Rightarrow$n = 2