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(A, B, C, D) For a charged particle in a magnetic field, the radius of the circular path is given by $r = \frac{mv}{qB}$.Applying Bohr's quantization

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Transition frequency $\omega$ between two successive levels is independent of $n$

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(A, B, C, D) For a charged particle in a magnetic field, the radius of the circular path is given by $r = \frac{mv}{qB}$.Applying Bohr's quantization condition for angular momentum: $mvr = \frac{nh}{2\pi} = n\hbar$.Substituting $v = \frac{qBr}{m}$ into the quantization condition: $m \left( \frac{qBr}{m} ight) r = n\hbar \implies qBr^2 = n\hbar \implies r_n = \sqrt{\frac{n\hbar}{qB}}$. Thus, $r_n \propto \sqrt{n}$. (Statement $A$ is true).Now, $v_n = \frac{qBr_n}{m} = \frac{qB}{m} \sqrt{\frac{n\hbar}{qB}} = \sqrt{\frac{n q B \hbar}{m^2}}$. (Statement $B$ is true).The energy of the $n^{	ext{th}}$ level is $E_n = \frac{1}{2}mv_n^2 = \frac{1}{2}m \left( \frac{n q B \hbar}{m^2} ight) = n \left( \frac{q B \hbar}{2m} ight)$. Thus, $E_n \propto n$. (Statement $C$ is true).The transition frequency $\omega$ is given by $\Delta E = \hbar \omega$. For successive levels, $\Delta E = E_{n+1} - E_n = \frac{q B \hbar}{2m}$. Therefore, $\omega = \frac{\Delta E}{\hbar} = \frac{qB}{2m}$, which is independent of $n$. (Statement $D$ is true).

$A$ uniform magnetic field $B$ exists in a region. An electron of charge $q$ and mass $m$ moving with velocity $v$ enters the region in a direction perpendicular to the magnetic field. Considering Bohr angular momentum quantization, which of the following statement$(s)$ is/are true?

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