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)

The correct answer is:
$\frac{he}{2\pi m}$

Step-by-step derivation:

1. Magnetic Flux Quantization:
Let the electron revolve in a circular orbit of radius r perpendicular to the uniform magnetic field B.
The area A of this circular orbit is:
$A=\pi r^2$
The magnetic flux passing through this orbit is:
$\Phi =B\cdot A=B\left(\pi r^2\right)$
According to the model, the magnetic flux is quantized as:
$\Phi =n\frac{h}{e}$
Equating the two expressions for flux, we get:
$B\left(\pi r^2\right)=\frac{nh}{e}$
From this, we express r² as:
$r^2=\frac{nh}{\pi eB}$ ---- (Equation 1)

2. Centripetal Force and Velocity:
For an electron of mass m and charge magnitude e moving in a circle of radius r with speed v, the magnetic force provides the necessary centripetal force:
$\frac{m{v}^2}{r}=evB$
Simplifying this relation, we find the orbital speed v:
$v=\frac{eBr}{m}$

3. Equivalent Orbital Current:
The time period T of the orbital motion is:
$T=\frac{2\pi r}{v}$
The equivalent electric current I due to the orbiting electron is:
$I=\frac{e}{T}=\frac{ev}{2\pi r}$
Substituting the expression for v into the current equation:
$I=\frac{e}{}$
2πr ⋅ eBr m = e2B 2πm

4. Magnetic Moment of the Electron:
The magnetic moment $\mu$ of the circular current loop is:
$\mu =I\cdot A=I\left(\pi r^2\right)$
Substituting the expressions for I and r² (from Equation 1):

Simplifying the terms:
$\mu =\frac{nhe}{2\pi m}$

5. Lowest Energy State:
For the lowest energy state, we consider the minimum quantum number, which is $n=1$:

$\mu =\frac{he}{2\pi m}$