Question

Define magnetic moment and write its unit. An electron is moving with the velocity \( 0 \times 10^7 \, \text{ms}^{-1} \) in a circular orbit of radius 0.3 \AA. Calculate its magnetic moment.

Hint:

The magnetic moment for a moving charge in a circular orbit is given by \( \mu = m \cdot v \cdot r \), where \( m \) is the mass, \( v \) is the velocity, and \( r \) is the radius of the orbit.

Answer 1

Step 1: Definition of Magnetic Moment.
Magnetic moment (\( \mu \)) is the measure of the strength and orientation of a magnetic source. For a moving charge, it is given by the formula: \[ \mu = I \cdot A \] Where:
- \( I \) is the current,
- \( A \) is the area enclosed by the moving charge.
For an electron moving in a circular path, we can express the magnetic moment as: \[ \mu = m \cdot v \cdot r \] Where:
- \( m \) is the mass of the electron (\( m_e = 9.11 \times 10^{-31} \, \text{kg} \)),
- \( v \) is the velocity of the electron (\( 0 \times 10^7 \, \text{ms}^{-1} \)),
- \( r \) is the radius of the circular orbit (\( 0.3 \, \text{Å} = 0.3 \times 10^{-10} \, \text{m} \)).
Step 2: Calculation of Magnetic Moment.
Substitute the values into the formula for magnetic moment: \[ \mu = 9.11 \times 10^{-31} \times 0 \times 10^7 \times 0.3 \times 10^{-10} \] \[ \mu = 5.46 \times 10^{-34} \, \text{A·m}^2 \]
Step 3: Unit of Magnetic Moment.
The SI unit of magnetic moment is \( \text{A·m}^2 \) (ampere square meter).
Final Answer:
The magnetic moment of the electron is \( \boxed{5.46 \times 10^{-34} \, \text{A·m}^2} \).