From the following, the quantity ( constructed from the basic con… - Vidyadip

MCQ

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

✓
$\frac{{{e^2}}}{{4\pi {\varepsilon _0}m{c^2}}}$
B
$\frac{{4\pi {\varepsilon _0}{e^2}}}{{m{e^2}}}$
C
$\frac{{m{e^2}}}{{4\pi {\varepsilon _0}{b^2}}}$
D
$\frac{{4\pi {\varepsilon _0}m{c^2}}}{{{e^2}}}$

✓

Answer

Correct option: A.

$\frac{{{e^2}}}{{4\pi {\varepsilon _0}m{c^2}}}$

a
As $\mathrm{E}=\mathrm{mc}^{2}$

also $\mathrm{E}=\mathrm{F} \cdot \mathrm{s}=\frac{\mathrm{k} q \cdot \mathrm{q}}{\mathrm{r}^{2}} \cdot \mathrm{r}$

Therefore dimensionally

$\mathrm{mc}^{2}=\frac{1}{4 \pi \in_{0}} \frac{\mathrm{q}^{2}}{\mathrm{r}}$

$\Rightarrow r=\frac{e^{2}}{4 \pi \in_{0} m c^{2}}$

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