Explanation

The dimension of $y = \frac{e^{2}}{4\pi\varepsilon_{0}hc}$ Putting the dimensions of $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lbrack e\rbrack = \lbrack Q\rbrack = \lbrack AT\rbrack$ $\ \ \ \ \left\lbrack \varepsilon_{0} \right\rbrack = \left\lbrack M^{- 1}L^{- 3}T^{4}A^{2} \right\rbrack,\ h = \left\lbrack ML^{2}T^{- 1} \right\rbrack,\ c = \left\lbrack LT^{- 1} \right\rbrack$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y = \frac{\left\lbrack A^{2}T^{2} \right\rbrack}{\left\lbrack M^{- 1}L^{- 3}T^{4}A^{2} \right\rbrack\left\lbrack ML^{2}T^{- 1} \right\rbrack\left\lbrack LT^{- 1} \right\rbrack}$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y = \left\lbrack M^{0}L^{0}T^{0} \right\rbrack$