If $$e$$ is the electronic charge, $$c$$ is the speed of light in free space and $$h$$ is Planck's constant, the quantity $$\frac{1}{4\pi\varepsilon_0} \frac{|e|^2}{hc}$$ has dimensions of:

We need to find the dimensions of the quantity $$\frac{1}{4\pi\varepsilon_0} \frac{|e|^2}{hc}$$.

This expression is the well-known fine structure constant $$\alpha$$. Let us verify it is dimensionless by checking the dimensions of each quantity involved.

The quantity $$\frac{1}{4\pi\varepsilon_0} \frac{e^2}{r}$$ represents electrostatic potential energy, which has dimensions of energy $$[ML^2T^{-2}]$$. Therefore $$\frac{1}{4\pi\varepsilon_0} e^2$$ has dimensions $$[ML^3T^{-2}]$$.

Now, Planck's constant $$h$$ has dimensions $$[ML^2T^{-1}]$$ and the speed of light $$c$$ has dimensions $$[LT^{-1}]$$. So the product $$hc$$ has dimensions $$[ML^2T^{-1}][LT^{-1}] = [ML^3T^{-2}]$$.

Therefore, the ratio $$\frac{1}{4\pi\varepsilon_0} \frac{|e|^2}{hc}$$ has dimensions $$\frac{[ML^3T^{-2}]}{[ML^3T^{-2}]} = [M^0L^0T^0]$$.

The quantity is dimensionless, and its numerical value is approximately $$\frac{1}{137}$$, known as the fine structure constant.