From the following, the quantity (constructed from the basic...

From the following, the quantity (constructed from the basic constants of nature), that has the dimensions, as well as correct order of magnitude, vis-a-vis typical atomic size, is:

Options

A $\frac{e^2}{4 \pi \varepsilon_0 m c^2}$
B $\frac{4 \pi \varepsilon_0 e^2}{m e^2}$
C $\frac{m e^2}{4 \pi \varepsilon_0 b^2}$
D $\frac{4 \pi \varepsilon_0 m c^2}{e^2}$

Correct answer

A. $\frac{e^2}{4 \pi \varepsilon_0 m c^2}$

Step-by-step solution

\mathrm{E}=\mathrm{mc}^2

also

\mathrm{E}=\mathrm{F} \cdot \mathrm{s}=\frac{\mathrm{kq} \cdot \mathrm{q}}{\mathrm{r}^2} \cdot \mathrm{r}

Therefore dimensionally

\begin{aligned} & \mathrm{mc}^2=\frac{1}{4 \pi \epsilon_0} \frac{\mathrm{q}^2}{\mathrm{r}} \\ & \Rightarrow \mathrm{r}=\frac{\mathrm{e}^2}{4 \pi \epsilon_0 \mathrm{mc}^2} \end{aligned}

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