Question

A uniform magnetic field $B$ exists in a region. An electron of charge $e$ is given velocity perpendicular to the magnetic field. Using, Bohr's quantization rule for angular momentum, the radius of the $n^{th}$ orbit is $\frac{nh}{k\pi eB}$, here $h$ is plank's constant. The value of $k$ is (Integer only)

Answer 1

The required centripetal force is provided by the magnetic Lorentz force,

$\Rightarrow evB=\frac{m{v}^2}{r}$

$\Rightarrow r=\frac{mv}{eB}---\left(1\right)$

From Bohr's angular momentum quantization rule,

$mvr=\frac{nh}{2\pi }$

$\Rightarrow mv=\frac{nh}{2\pi r}---\left(2\right)$

Substituting $\left(2\right)$ in $\left(1\right)$,

$\Rightarrow r^2=\frac{nh}{2\pi e}\frac{1}{B}$

$\Rightarrow r_n=\sqrt{\frac{nh}{2\pi eB}}$

Given,

$r_n=\sqrt{\frac{nh}{k\pi eB}}$

$\therefore k=2$