MHT CET · Physics · Atomic Physics

Using Bohr's model, the orbital period of electron in hydrogen atom in \(\mathrm{n}^{\text {th }}\) orbit is ( \(\mathrm{m}=\) mass of electron, \(\mathrm{h}=\) Planck's constant, \(\mathrm{e}=\) electronic charge, \(\varepsilon_0=\) permittivity of free space)

A \(\frac{2 \varepsilon_0^2 \mathrm{n}^2 \mathrm{~h}^2}{\mathrm{me}^4}\)
B \(\frac{4 \varepsilon_0^2 \mathrm{n}^2 \mathrm{~h}^2}{\mathrm{me}^2}\)
C \(\frac{4 \varepsilon_0^2 \mathrm{n}^3 \mathrm{~h}^3}{m \mathrm{e}^4}\)
D \(\frac{4 \varepsilon_0 \mathrm{n}^2 \mathrm{~h}^2}{\pi \mathrm{me}^2}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{4 \varepsilon_0^2 \mathrm{n}^3 \mathrm{~h}^3}{m \mathrm{e}^4}\)

Step-by-step Solution

Detailed explanation

\(\mathrm{r}=\) radius of \(\mathrm{n}^{\text {th }}\) orbit \(=\frac{\mathrm{n}^2 \mathrm{~h}^2}{\pi \mathrm{mZe}^2}\)
\(\mathrm{v}=\) speed of \(\mathrm{e}^{-}\)in \(\mathrm{n}^{\text {th }}\) orbit \(=\frac{\mathrm{Ze}^2}{2 \varepsilon_0 n h}\)
\(\mathrm{T}=\frac{2 \pi \mathrm{r}}{\mathrm{v}}\)
Substituting values of v and r ,
\(\therefore \quad \mathrm{T}=\frac{4 \varepsilon_0^2 \mathrm{n}^3 \mathrm{~h}^3}{m \mathrm{Z}^2 \mathrm{e}^4}\)
For hydrogen, \(\mathrm{Z}=1\)
\(\therefore \quad T=\frac{4 \varepsilon_0^2 \mathrm{n}^3 \mathrm{~h}^3}{\mathrm{me}^4}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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