A model for quantized motion of an electron in a uniform magnetic...

A model for quantized motion of an electron in a uniform magnetic field

B

states that the flux passing through the orbit of the electron is

n(h / e)

where

n

is an integer,

h

is Planck's constant and

e

is the magnitude of electron's charge. According to the model, the magnetic moment of an electron in its lowest energy state will be (

m

is the mass of the electron)

Options

B. $\frac{h e}{2 \pi m}$

Magnetic moment

\begin{aligned} & M=I A=I\left(\pi r^2\right) \\ & M=\left(\frac{e \mathrm{u}}{2 \pi r}\right)\left(\pi r^2\right) \end{aligned}

Given

B\left(\pi r^2\right)=n\left(\frac{h}{e}\right) \Rightarrow \mathrm{r}^2=\frac{\mathrm{h}}{\mathrm{~B} \pi \mathrm{e}} \quad \ldots(2) \quad(\because \mathrm{n}=1)

And when charge moving in external magnetic field then

r=\frac{m u}{q B} \frac{\mathrm{v}}{\mathrm{r}}=\frac{e \mathrm{~B}}{\mathrm{~m}} \ldots(3) \quad(\because \mathrm{q}=e)

Put value from equation (2) and (3) in equation (1)

\begin{aligned} & M=\left(\frac{e v}{2 \pi r}\right)\left(\pi r^2\right) \\ & M=\frac{e}{2 \pi}\left(\frac{e B}{m}\right) \pi\left(\frac{h}{B \pi e}\right) \\ & M=\frac{e h}{2 \pi m} \end{aligned}