An electron is revolving around a proton in an orbit of radius r with a…

An electron is revolving around a proton in an orbit of radius r with a speed v. Obtain expression for magnetic moment associated with the electron.

Ans. The magnetic moment associated with the electron in its orbit around the proton can be calculated using the formula for the magnetic moment of a current loop.

The current in the loop is due to the moving electron. The current, $I$, is the charge $q$ of the electron moving in a circle of radius $r$ with velocity $v$. So, we have:

�=��I=Tq​

Where $T$ is the time period of the orbit.

The time period of the orbit is the time taken by the electron to complete one revolution around the proton, which can be given by:

�=2���T=v2πr​

Now, substituting $T$ in the expression for $I$, we get:

�=�2���=��2��I=v2πr​q​=2πrqv​

The magnetic moment $\mu$ of the current loop is given by the product of the current $I$ and the area $A$ of the loop, which is equal to ��2πr2:

$\mu =I\cdot A$

�=��2��⋅��2=12���μ=2πrqv​⋅πr2=21​qvr

So, the expression for the magnetic moment associated with the electron is 12���21​qvr.