Question

If one were to apply Bohr model to a particle “m” and charge “q” moving in a plane under the influence of the magnetic field “B”, the energy of the charged particle in the nth level will be:
(A) $n\left( \dfrac{hqB}{4\pi m} \right)$
(B) $n\left( \dfrac{hqB}{8\pi m} \right)$
(C) $n\left( \dfrac{hqB}{\pi m} \right)$
(D) $n\left( \dfrac{hqB}{2\pi m} \right)$

Answer 1

Here we will apply the Bohr quantization condition. The value of $vr$ can be calculated from here.
The magnetic Lorentz force provides the necessary centripetal force. From here, value of $\dfrac{v}{r}$ can be calculated
By multiplying these two values found, we can find the value of ${{v}^{2}}$ and hence kinetic energy using the formula $\dfrac{1}{2}m{{v}^{2}}$.

Complete step by step solution
Bohr’s quantization condition states that angular momentum of an electron is an integral multiple of $\dfrac{h}{2\pi }$
$\text{L=}\dfrac{n\text{ }h}{2\pi }$
As $\alpha =mvr$ , so
$mvr=\dfrac{nh}{2\pi }$
$vr=\dfrac{nh}{2\pi m}$ …… …… …… (1)
Now, the magnetic Lorentz force provides the necessary centripetal force.
So, $\dfrac{m{{v}^{2}}}{r}=qvB$
On rearranging, we get
$\dfrac{v}{r}=\dfrac{qB}{m}$ …… …… …… (2)
On multiplying equations (1) and (2) we get
$vr\times \dfrac{v}{r}=\left( \dfrac{nh}{2\pi m} \right)\times \left( \dfrac{qB}{m} \right)$
${{v}^{2}}=\dfrac{n\text{ }h\ q\text{ }B}{2\pi \text{ }{{m}^{2}}}$
Now energy of the charged particle is given by:
K.E $=\dfrac{1}{2}m{{v}^{2}}$
$=\dfrac{1}{2}m\left( \dfrac{n\text{ }h\text{ }q\text{ }B}{2\pi {{m}^{2}}} \right)$
$\text{E=n}\left( \dfrac{hqB}{4\pi m} \right)$.
Correct option is (A).

Note
Lorentz force is the force exerted on a charged particle q moving with velocity v through an electric field E and moving magnetic field B.
The entire electromagnetic force F on the charged particle is called the Lorentz force and is given by
$\text{F=}qE+qv\times B$
The first term is contributed by the electric field. The second term is the magnetic field. The second term is the magnetic field and has a direction perpendicular to both the velocity and the magnetic field.