The magnetic moment (m_{orb}) of a revolving | vedclass

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(D) The orbital magnetic moment is defined as $m_{orb} = iA$,where $i = \frac{e}{T}$ is the current,$A = \pi r^2$ is the area,and $T$ is the time peri

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$m_{orb} \propto n$

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(D) The orbital magnetic moment is defined as $m_{orb} = iA$,where $i = \frac{e}{T}$ is the current,$A = \pi r^2$ is the area,and $T$ is the time period of the electron's revolution.Substituting $i$ and $A$,we get $m_{orb} = e \left( \frac{\pi r^2}{T} \right) \dots (1)$.According to Bohr's quantization condition for angular momentum,$mvr = \frac{nh}{2\pi} \dots (2)$.Also,the time period $T$ is related to velocity $v$ and radius $r$ by $T = \frac{2\pi r}{v}$,which implies $\frac{r}{T} = \frac{v}{2\pi}$.Substituting this into the expression for $m_{orb}$,we get $m_{orb} = e \pi r \left( \frac{r}{T} \right) = e \pi r \left( \frac{v}{2\pi} \right) = \frac{evr}{2}$.From equation $(2)$,$vr = \frac{nh}{2\pi m}$.Substituting this into the expression for $m_{orb}$,we get $m_{orb} = \frac{e}{2} \left( \frac{nh}{2\pi m} \right) = n \left( \frac{eh}{4\pi m} \right)$.Since $e, h, \pi,$ and $m$ are constants,we conclude that $m_{orb} \propto n$.

The magnetic moment $(m_{orb})$ of a revolving electron around the nucleus varies with the principal quantum number $(n)$ as

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