Question

Calculate the angle created by orbital magnetic moment with the angular momentum of the electron?
$\begin{align}
  & A.120{}^\circ \\
 & B.60{}^\circ \\
 & C.180{}^\circ \\
 & D.90{}^\circ \\
\end{align}$

Answer 1

Hint: The time period of the electron revolving in a circular orbit has been given as the ratio of the circumference of the orbit to the velocity of travel. Current created due to the motion of electrons is given as the ratio of the charge of the electron to the time period. The magnetic moment due to the current loop will be the product of current and the area of cross section. This will help you in answering this question.

Complete answer:
 Let us assume that an electron is revolving in a circle of radius \[r\] with a velocity \[v\]. Let the charge of the electron is \[e\] and its mass is \[m\]. Both of them are fixed values. The time period \[T\] of the orbit of the electron is,
\[T=\dfrac{circumference}{velocity}=\dfrac{2\pi r}{v}\]
The current \[i\] because of the motion of the electron will be the charge flowing through that time period.
That is,
\[i=\dfrac{e}{\dfrac{2\pi r}{v}}=\dfrac{ev}{2\pi r}\]
The current will be in the opposite direction as the electron is negatively charged.
Magnetic moment because of a current loop enclosing an area \[A\] can be written as,
\[\mu =iA\]
The magnetic moment of the electron is,
\[\begin{align}
  & \mu =-\dfrac{ev}{2\pi r}A=-\dfrac{ev}{2\pi r}\times \pi {{r}^{2}} \\
 & \Rightarrow \mu =-\dfrac{erv}{2} \\
\end{align}\]
Now we can divide and multiply the equation with mass \[m\],
\[\mu =-\dfrac{e}{2m}mvr\]
The angular momentum can be written as,
\[L=mvr\]
That is,
\[\mu =-\dfrac{e}{2m}L\]
As the angular momentum is found by the right-hand rule. That is the velocity and the current in the opposite direction. Therefore, the negative sign shows that the two quantities are in opposing directions.
\[\mu =-\dfrac{e}{2m}L\]
It is to be noted that the magnetic moment is only dependent upon the angular momentum. This is the reason for the orbital angular momentum and orbital magnetic moment terms can be interchangeably used. This will be true for the spin angular moment.

So, the correct answer is “Option C”.

Note:
The magnetic moment is defined as the measure of the tendency of a magnet to get aligned according to a magnetic field. A magnet will have two poles. They are the North Pole and South Pole. A loop of electric current is an example for an object with a magnetic moment.