question_answer

An electron of charge e moves in a circular orbit of radius r around the nucleus at a frequency v. The magnetic moment associated with the orbital motion of the electron is

A)  \[\pi ve{{r}^{2}}\]                          

B) \[\frac{\pi v{{r}^{2}}}{e}\]

C)  \[\frac{\pi v\,e}{r}\]                      

D)        \[\frac{\pi v{{r}^{2}}}{v}\]

Correct Answer: A

Solution :

Magnetic moment or magnetic dipole moment is a measure of the strength of a magnetic source. It is given by                 \[\mu =IA\]                                                        ? (i) where \[I\]is current and\[A\]the area. The effective current\[(I)\]is\[I=\frac{e}{T}\] where\[e\]is electron charge and\[T\]the time period. Also,\[v=\frac{1}{T}=\]frequency and area \[A=\pi {{r}^{2}}\], where\[r\]is radius of circular path. Putting these values in Eq. (i), we get                 \[\mu =ev.\pi {{r}^{2}}\] Note: Magnetic moment can also expressed terms of angular momentum\[(L)\] \[\mu =IA=-\frac{2}{2{{M}_{e}}}L\]. where\[{{M}_{e}}\] is mass of electron.