MHT CET · Physics · Atomic Physics
Bohr model is applied to a particle of mass ' \(\mathrm{m}\) ' and charge ' \(\mathrm{q}\) ' moving in a plane under the influence of a transverse magnetic field ' \(\mathrm{B}\) '. The energy of the charged particle in the \(\mathrm{n}^{\text {th }}\) level will be \([\mathrm{h}=\) Planck's constant \(]\)
- A \(\frac{\mathrm{nhqB}}{4 \pi \mathrm{m}}\)
- B \(\frac{\mathrm{nhqB}}{2 \pi \mathrm{m}}\)
- C \(\frac{\mathrm{nhqB}}{\pi \mathrm{m}}\)
- D \(\frac{2 \mathrm{nhqB}}{\pi \mathrm{m}}\)
Answer & Solution
Correct Answer
(A) \(\frac{\mathrm{nhqB}}{4 \pi \mathrm{m}}\)
Step-by-step Solution
Detailed explanation
We know,
\(\mathrm{mvr} =\frac{\mathrm{nh}}{2 \pi} \)
\( \therefore \mathrm{vr} =\frac{\mathrm{nh}}{2 \pi \mathrm{m}}... (i)\)
Also,
\(\mathrm{qvB} =\frac{\mathrm{mv}^2}{\mathrm{r}} \)
\( \therefore \mathrm{mv} =\mathrm{qBr} ... (ii)\)
\( \mathrm{mv}^2 \mathrm{r}=\mathrm{qBr} \times \frac{\mathrm{nh}}{2 \pi \mathrm{m}} \ldots (\text { Multiplying (i) with (ii)) } \)
\( \mathrm{E} =\frac{1}{2} \mathrm{mv}^2=\mathrm{n}\left[\frac{\mathrm{qBh}}{4 \pi \mathrm{m}}\right]\)
\(\mathrm{mvr} =\frac{\mathrm{nh}}{2 \pi} \)
\( \therefore \mathrm{vr} =\frac{\mathrm{nh}}{2 \pi \mathrm{m}}... (i)\)
Also,
\(\mathrm{qvB} =\frac{\mathrm{mv}^2}{\mathrm{r}} \)
\( \therefore \mathrm{mv} =\mathrm{qBr} ... (ii)\)
\( \mathrm{mv}^2 \mathrm{r}=\mathrm{qBr} \times \frac{\mathrm{nh}}{2 \pi \mathrm{m}} \ldots (\text { Multiplying (i) with (ii)) } \)
\( \mathrm{E} =\frac{1}{2} \mathrm{mv}^2=\mathrm{n}\left[\frac{\mathrm{qBh}}{4 \pi \mathrm{m}}\right]\)
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