Question
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 $n(hle)$ 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)
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Solution
Magnetic force $= \frac{mv^{2}}{r}$
$$\begin{matrix} & evB = \frac{mv^{2}}{r} \\ & v = \frac{eBr}{m} \\ & \phi = BA \\ & \frac{nh}{e} = B\pi r^{2} \\ & Br^{2} = \frac{nh}{e\pi} \\ & \mu = IA \\ & \ = \frac{e}{T}\pi r^{2} \\ & \ = \frac{e \times v}{2\pi r}\pi r^{2} \\ & \mu = \frac{evr}{2} \\ & \ = \frac{1}{2}e \times \frac{eBr}{m}r \\ & \mu = \frac{1}{2}e^{2}\frac{Br^{2}}{m} \\ & \mu = \frac{1}{2}e^{2}\frac{nh}{e\pi m} \\ & \mu = \frac{neh}{2\pi m} \end{matrix}$$
for $n = 1$
$$\mu = \frac{eh}{2\pi m}$$
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