Question

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

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

Answer 1

Answer: (C)

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