The ratio of speed of an electron in ground state in Bohr's first orbit of hydrogen atom to velocity of light in air is

$\frac{e^2}{2 \varepsilon_0 h c}$

Speed of electron in n^th orbit (in CGS) $v_n=\frac{2 \pi Z e^2}{n h}(k=1)$

For first orbit H₂ ; n = 1 and Z = 1

So $v=\frac{2 \pi e^2}{h} \Rightarrow \frac{v}{c}=\frac{2 \pi e^2}{h c}$