MHT CET · Physics · Magnetic Effects of Current

In the Bohr model, an electron moves in a circular orbit around the nucleus. Considering an orbiting electron to be a circular current loop, the magnetic moment of the hydrogen atom, when the electron is in nth excited state, is ( \(\mathrm{e}=\) electronic charge, \(\mathrm{m}_{\mathrm{e}}=\) mass of the electron, \(\mathrm{h}=\) Planck's constant)

A \(\left(\frac{\mathrm{e}}{\mathrm{m}_{\mathrm{e}}}\right) \frac{\mathrm{nh}}{2 \pi}\)
B \(\left(\frac{\mathrm{e}}{\mathrm{m}_{\mathrm{e}}}\right) \frac{\mathrm{n}^2 \mathrm{~h}}{2 \pi}\)
C \(\left(\frac{\mathrm{e}}{2 \mathrm{~m}_{\mathrm{e}}}\right) \frac{\mathrm{n}^2 \mathrm{~h}}{2 \pi}\)
D \(\left(\frac{\mathrm{e}}{2 \mathrm{~m}_{\mathrm{e}}}\right) \frac{\mathrm{nh}}{2 \pi}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\left(\frac{\mathrm{e}}{2 \mathrm{~m}_{\mathrm{e}}}\right) \frac{\mathrm{nh}}{2 \pi}\)

Step-by-step Solution

Detailed explanation

Current due to motion of electron, \(i=\frac{e v}{2 \pi r}\) Magnetic moment,
\(
\mathrm{M}=\mathrm{iA}=\frac{\mathrm{ev}}{2 \pi \mathrm{r}} \times \pi \mathrm{r}^2=\frac{\mathrm{evr}}{2}=\frac{\mathrm{e}}{2 \mathrm{~m}}(\mathrm{mvr})=\frac{\mathrm{e}}{2 \mathrm{~m}}\left(\frac{\mathrm{nh}}{2 \pi}\right)
\)

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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