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

An electron orbiting the nucleus in a circular path generates a loop current.

The current \( I \) is defined as:

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

With the electron's charge being \( e \) and its time period \( T \):

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

Consequently,

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

Step 2: Magnetic Moment Calculation

The magnetic moment \( \mu \) is calculated using the formula:

\[ \mu = I \times A \]

For a circular orbit, the area \( A \) is:

\[ A = \pi r^2 \]

Therefore,

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

This simplifies to:

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

Step 3: Expressing in Terms of Angular Momentum

The angular momentum \( L \) of the electron is given by:

\[ L = mvr \]

where \( m \) is the electron's mass.

Using the previously derived equation:

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

Substituting \( vr \) from the angular momentum equation:

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

Thus, the magnetic moment associated with the orbiting electron is:

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