Bohr model is applied to a particle of mass 'm' and charge 'q' moving in a plane under the influence of a transverse magnetic field B. The energy of the charged particle in the n t

Question

Bohr model is applied to a particle of mass 'm' and charge 'q' moving in a plane under the influence of a transverse magnetic field B. The energy of the charged particle in the n t h level will be (h=Planck's constant)

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For a charged particle moving in a transverse magnetic field, the magnetic force provides the necessary centripetal force for circular motion:

$\frac{m v ^{2}}{r} = q v B$

$\Rightarrow v = \frac{qB r}{m}$

According to Bohr's quantization postulate, the angular momentum of the particle is quantized:

$m v r = \frac{nh}{2 π}$

Substituting the expression for $v$ into the quantization condition, we get:

$m (\frac{qB r}{m}) r = \frac{nh}{2 π}$

$\Rightarrow qB r^{2} = \frac{nh}{2 π}$

$\Rightarrow r^{2} = \frac{nh}{2 π qB}$

The total energy of the particle is purely kinetic (since the magnetic force does no work):

$E = \frac{1}{2} m v^{2}$

Substituting $v = \frac{qB r}{m}$ :

$E = \frac{1}{2} m (\frac{qB r}{m})^{2} = \frac{q ^{2} B ^{2} r ^{2}}{2 m}$

Now, substituting the value of $r^{2}$ into the energy equation:

$E = \frac{q ^{2} B ^{2}}{2 m} (\frac{nh}{2 π qB})$

$E = \frac{nh qB}{4 πm}$

Answer: $\frac{nh qB}{4 πm}$

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