A model for quantized motion of an electron in a uniform magnetic field B states that the flux passing through the orbit of the e…

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(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. A. he/π m
2. B. he/2π m (Correct answer)
3. C. heB/π m
4. D. heB/2π m

Correct Answer

Option B — he/2π m

Detailed Solution & Explanation

The correct answer is $\dfrac{he}{2\pi m}$.

Key Points

• The magnetic force provides the centripetal force: $evB=\dfrac{mv^2}{r}\Rightarrow v=\dfrac{eBr}{m}$.
• Quantized flux: $\phi=BA=B\pi r^2=\dfrac{nh}{e}\Rightarrow Br^2=\dfrac{nh}{e\pi}$.
• Magnetic moment $\mu=IA=\dfrac{e}{T}\pi r^2=\dfrac{evr}{2}=\dfrac{1}{2}e\cdot\dfrac{eBr}{m}\cdot r=\dfrac{e^2Br^2}{2m}$.
• Substituting $Br^2$: $\mu=\dfrac{e^2}{2m}\cdot\dfrac{nh}{e\pi}=\dfrac{neh}{2\pi m}$. For the lowest state $n=1$: $\mu=\dfrac{eh}{2\pi m}$.