Question

Bohr model is applied to a particle of mass $'m'$ and charge $'q'$ is moving in a plane under the influence of a transverse magnetic field $'B'$. The energy of the charged particle in the $nth$ level will be ($h =$ Planck's constant)

• A. $2nhq\,B/\pi\, m$
• B. $nhq\,B/2\pi\, m$
• C. $nhq\,B/4\pi\, m$
• D. $nhq\,B/\pi\, m$

Answer 1

Answer: (C)

$nhq\,B/4\pi\, m$

Answer 2

For a particle moving in a magnetic field, then applied two forces are equal.
centripetal force $\left(F_{c}\right)=$ magnetic force $\left(F_{m}\right)$
$\Rightarrow \frac{m v^{2}}{r}=q v B$
$\Rightarrow m v^{2}=q B(v r)$...(i)
Also, from Bohr's model,
$m v r=\frac{n h}{2 \pi}$
$\therefore v r=\frac{n h}{2 \pi m}$...(ii)
From E (i) and (ii), we get
$m v^{2}=\frac{n h}{2 \pi m} \cdot q B \dots$(iii)
Energy of the electron moving in $n$ h orbit,
$E =\frac{1}{2} \cdot m v^{2}=\frac{1}{2} \cdot \frac{n h q B}{2 \pi m}$(using E(iii)
$\Rightarrow E=\frac{n h q B}{4 \pi m}$
Hence, the energy of the charged particle in the nth level will be $\frac{n h q B}{4 \pi m}$.