Question

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

Answer 1

Magnetic Moment of an Orbiting Electron 

Step 1: Current Due to Electron Motion

The electron moves in a circular orbit around the nucleus, creating a loop current.

The current \( I \) is:

\[ I = \frac{\text{Charge per revolution}}{\text{Time for one revolution}} \]

The charge of an electron is \( e \) and time period \( T \) is:

\[ T = \frac{2\pi r}{v} \]

Thus,

\[ I = \frac{e}{T} = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r} \]

Step 2: Magnetic Moment Calculation

The magnetic moment \( \mu \) is given by:

\[ \mu = I \times A \]

Since the electron follows a circular path, the area is:

\[ A = \pi r^2 \]

Thus,

\[ \mu = \frac{ev}{2\pi r} \times \pi r^2 \]

\[ \mu = \frac{evr}{2} \]

Step 3: Expressing in Terms of Angular Momentum

The angular momentum of the electron is:

\[ L = mvr \]

where \( m \) is the mass of the electron.

Since:

\[ \mu = \frac{evr}{2} \]

Replacing \( vr \) using \( L \):

\[ \mu = \frac{e}{2m} L \]

Thus, the magnetic moment associated with the electron is:

\[ \mu = \frac{e}{2m} L \]