Question

An electron is orbiting in a circular orbit of radius $r$ under the influence of a constant magnetic field of strength $B$. Assuming that Bohr's postulate regarding the quantization of angular momentum holds good for this electron, find
a. the allowed values of the radius $r$ of the orbit
b. the kinetic energy of the electron in orbit.
c. the potential energy of the interaction between the magnetic moment of the orbital current due to the electron moving in its orbit and the magnetic field $B$
d. the total energy of the allowed energy levels
e. the total magnetic flux due to the magnetic field $B$ passing through the $nth$ orbit (assume that the charge on the electron is $e$ and the mass of the electron is $m$).

Answer 1

For the circular motion of electron, we can write $\frac{m{v}^2}{r}=evB$

or $mv=eBr.....\left(1\right)$

From Bohr's postulate, $mvr=n\hslash ....\left(2\right)$ where n=interger

a). using (1), $\left(eBr\right)r=n\hslash \Rightarrow r=\sqrt{\frac{n\hslash }{eB}}$

b) The kinetic energy, $KE=\frac{1}{2}m{v}^2=\frac{1}{2}m\times \frac{n^2{\hslash }^2}{m^2{r}^2}=\frac{1}{2m}\times \frac{n^2{\hslash }^{2}eB}{n\hslash }=\frac{n\hslash eB}{2m}$

c) Time period, $T=\frac{2\pi r}{v}=\frac{2\pi m}{eB}$ using (1)

The current, $I=\frac{e}{T}=\frac{e^{2}B}{2\pi m}$

Magnetic moment, $\mu =I\times area=\frac{e^{2}B}{2\pi m}\times \pi r^2=\frac{e^{2}B}{2m}\times \frac{n\hslash }{eB}=\frac{en\hslash }{2m}$

The potential energy, $PE=-\overrightarrow{\mu }.\overrightarrow{B}=\frac{en\hslash B}{2m}$

d) Total energy $=KE+PE=\frac{en\hslash }{m}$

e) Total magnetic flux, $\varphi =B.\pi r^2=B\pi \left(\frac{n\hslash }{eB}\right)=\frac{\pi n\hslash }{e}$