What is the magnetic moment of an orbiting el | vedclass

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(B) When an electron orbits around the hydrogen nucleus,it constitutes a current loop.The current $i$ is given by $i = \frac{e}{T}$,where $T$ is the t

Vedclass · Accepted Answer

$\vec{\mu}=-\left(\frac{e}{2 m_e}\right) \vec{L}$

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(B) When an electron orbits around the hydrogen nucleus,it constitutes a current loop.The current $i$ is given by $i = \frac{e}{T}$,where $T$ is the time period of revolution.The magnetic moment is $\mu = i A = \frac{e}{T} (\pi r^2)$.The time period $T = \frac{2 \pi r}{v}$. Substituting this into the expression for $\mu$:$\mu = \frac{e v}{2 \pi r} (\pi r^2) = \frac{e v r}{2}$.The orbital angular momentum is $L = m_e v r$,which implies $v r = \frac{L}{m_e}$.Substituting $v r$ into the expression for $\mu$:$\mu = \frac{e}{2} \left(\frac{L}{m_e}\right) = \frac{e}{2 m_e} L$.Since the electron is negatively charged,the magnetic moment vector $\vec{\mu}$ and angular momentum vector $\vec{L}$ are in opposite directions:$\vec{\mu} = -\left(\frac{e}{2 m_e}\right) \vec{L}$.

What is the magnetic moment of an orbiting electron in a simple hydrogen atom? Assume $e=$ charge of electron,$m_e=$ mass of electron,and $\vec{L}=$ orbital angular momentum of the electron.

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