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Question

(B) In the Bohr model,the velocity of an electron in the $n^{th}$ orbit is given by $v_n = \frac{Z e^2}{2 \epsilon_0 n h}$.In $CGS$ units,the Coulomb

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$\frac{2 \pi e^2}{hc}$

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(B) In the Bohr model,the velocity of an electron in the $n^{th}$ orbit is given by $v_n = \frac{Z e^2}{2 \epsilon_0 n h}$.In $CGS$ units,the Coulomb constant $k = \frac{1}{4 \pi \epsilon_0} = 1$,which implies $\epsilon_0 = \frac{1}{4 \pi}$.Substituting $\epsilon_0 = \frac{1}{4 \pi}$ into the velocity formula for $Z=1$ and $n=1$ (ground state):$v_1 = \frac{1 \cdot e^2}{2 (1/4 \pi) (1) h} = \frac{4 \pi e^2}{2 h} = \frac{2 \pi e^2}{h}$.The ratio of this velocity to the speed of light $c$ is $\frac{v_1}{c} = \frac{2 \pi e^2}{hc}$.

The ratio of the velocity of an electron in the ground state of a hydrogen atom to the velocity of light (in $CGS$ units) is:

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