Bohr model is applied to a particle of mass ' $\mathrm{m}$ ' and charge ' $\mathrm{q}$ ' moving in a plane…

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 $]$
  1. $\frac{\mathrm{nhqB}}{4 \pi \mathrm{m}}$
  2. $\frac{\mathrm{nhqB}}{2 \pi \mathrm{m}}$
  3. $\frac{\mathrm{nhqB}}{\pi \mathrm{m}}$
  4. $\frac{2 \mathrm{nhqB}}{\pi \mathrm{m}}$

Solution

We know, $\begin{aligned} \mathrm{mvr} & =\frac{\mathrm{nh}}{2 \pi} \\ \therefore \quad \mathrm{vr} & =\frac{\mathrm{nh}}{2 \pi \mathrm{m}}... (i) \end{aligned}$ Also, $\begin{aligned} \mathrm{qvB} & =\frac{\mathrm{mv}^2}{\mathrm{r}} \\ \therefore \quad \mathrm{mv} & =\mathrm{qBr} ... (ii)\\ & \mathrm{mv}^2 \mathrm{r}=\mathrm{qBr} \times \frac{\mathrm{nh}}{2 \pi \mathrm{m}} \quad \ldots . . .(\text { Multiplying (i) with (ii)) } \\ \mathrm{E} & =\frac{1}{2} \mathrm{mv}^2=\mathrm{n}\left[\frac{\mathrm{qBh}}{4 \pi \mathrm{m}}\right] \end{aligned}$ :

Asked in: MHT CET 2023 (11 May Shift 2)

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