Using Bohr's model, the orbital period of electron in hydrogen atom in $\mathrm{n}^{\text {th }}$ orbit is…

Using Bohr's model, the orbital period of electron in hydrogen atom in $\mathrm{n}^{\text {th }}$ orbit is $\left(\epsilon_{0}=\right.$ permittivity of free space, $\mathrm{h}=$ Planck's constant, $\mathrm{m}=$ mass of electron, $\mathrm{e}=$ electronic charge $)$
  1. $\frac{8 \epsilon_{0}^{2} \mathrm{n}^{3} \mathrm{~h}^{3}}{\mathrm{me}^{4}}$
  2. $\frac{2 \epsilon_{0}^{2} \mathrm{n}^{3} \mathrm{~h}^{3}}{\mathrm{~m} \mathrm{e}^{4}}$
  3. $\frac{2 \varepsilon_{0} \mathrm{n}^{2} \mathrm{~h}^{2}}{\mathrm{me}^{4}}$
  4. $\frac{4 \epsilon_{0}^{2} \mathrm{n}^{3} \mathrm{~h}^{3}}{\mathrm{me}^{4}}$

Solution

$\begin{array}{l} \mathrm{T}=\frac{2 \pi \mathrm{r}}{\mathrm{V}} ; \quad \mathrm{r}=\frac{\mathrm{n}^{2} \mathrm{~h}^{2} \epsilon_{0}}{\pi \mathrm{h} \mathrm{e}^{2}} \\ \mathrm{~V}=\frac{\mathrm{e}^{2}}{2 \pi \epsilon_{\mathrm{o}} \mathrm{n}} \end{array}$ putting the values of $\mathrm{V}$ and $\mathrm{r}$, $T=\frac{4 \epsilon_{0}{ }^{2} \mathrm{n}^{3} \mathrm{~h}^{3}}{\mathrm{me}^{4}}$

Asked in: MHT CET 2020 (19 Oct Shift 2)

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