Question

Drive an expression for the magnetic moment of an electron revolving around the nucleus in term of its angular momentum what is the direction of magnetic moment of the electron with respect to its angular momentum

Answer 1

Expression for the magnetic moment of the electron:

Let the charge of the electron be $\text{'}\mathrm{e}\text{'}$ and is revolving around a circular orbit of radius $\text{'}\mathrm{r}\text{'}$ with angular velocity $\text{'}\mathrm{\omega }\text{'}$.

The expression for the current can be given as: $\mathrm{I}=\frac{\mathrm{e}}{\mathrm{T}}$; where T is the time period of revolution which can be expressed as: $\mathrm{T}=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$

$\Rightarrow \mathrm{I}=\frac{\mathrm{e}}{{\displaystyle \frac{2\mathrm{\pi }}{\mathrm{\omega }}}}=\frac{\mathrm{e\omega }}{2\mathrm{\pi }}------\left(\mathrm{i}\right)$

We know that the magnetic moment$\left(\mathrm{M}\right)$ of Current is expressed as: $\mathrm{M}=\mathrm{IA}$ ; where I is current and A is the area of circular orbit expressed as: $\mathrm{A}={\mathrm{\pi r}}^2$

$\mathrm{M}=(-\frac{\mathrm{e}}{\mathrm{T}}){\mathrm{\pi r}}^2-----\left(\mathrm{ii}\right)$

The relation between linear velocity and angular velocity is given as: $\mathrm{v}=\mathrm{r\omega }$

Angular velocity can be expressed in terms of times period as: $\mathrm{\omega }=\frac{2\mathrm{\pi }}{\mathrm{T}}$

$\Rightarrow \mathrm{v}=\frac{2\mathrm{\pi r}}{\mathrm{T}}$

The angular moment is given as: $\mathrm{L}=\mathrm{mvr}$

Substituting value of ‘v’ in L

$\Rightarrow \mathrm{L}=\mathrm{m}\frac{2\mathrm{\pi r}}{\mathrm{T}}\mathrm{r}-----\left(\mathrm{iii}\right)$

Dividing eq(ii) by eq(iii)

$$\begin{gathered} \Rightarrow \frac{\mathrm{M}}{\mathrm{L}}=-\frac{\mathrm{e}}{\mathrm{T}}{\mathrm{\pi r}}^2\times \frac{\mathrm{T}}{2{\mathrm{\pi mr}}^2} \\ \Rightarrow \frac{\mathrm{M}}{\mathrm{L}}=-\frac{\mathrm{e}}{2\mathrm{m}} \\ \Rightarrow \overrightarrow{\underset{ }{M}}=(-\frac{\mathrm{e}}{2\mathrm{m}})\overrightarrow{\mathrm{L}} \end{gathered}$$

From the expression it can be inferred that direction of M and L will be opposite to each other.