Topic: Bohr model, energy levels, hydrogen spectrum· Atoms and Nuclei

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…

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)

1. $\frac{he}{\pi m}$

2. $\frac{he}{2\pi m}$

3. $\frac{heB}{\pi m}$

4. $\frac{heB}{2\pi m}$

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Answer: 2. $\frac{he}{2\pi m}$

Key idea: the electron moves in a circle in the field, and the given flux quantization fixes the orbit area.

From $evB=\dfrac{mv^2}{r}$: $v=\dfrac{eBr}{m}$, and the period is $T=\dfrac{2\pi m}{eB}$. Magnetic moment:

$$\mu=IA=\frac{e}{T}\,\pi r^2=\frac{e^2B}{2\pi m}\,\pi r^2=\frac{e^2}{2\pi m}\,(B\pi r^2)$$

Lowest state is $n=1$, so the flux $B\pi r^2=\dfrac he$:

$$\mu=\frac{e^2}{2\pi m}\cdot\frac he=\frac{he}{2\pi m}$$

The result does not contain $B$, so options 3 and 4 are out.

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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…

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