Question

Gyromagnetic ratio of an electron is the ratio between

• A. charge and angular momentum
• B. magnetic moment and angular acceleration
• C. magnetic moment and angular momentum
• D. charge and angular momentum
• E. magnetic moment and angular velocity

Hint:

Gyromagnetic ratio always connects magnetic moment with angular momentum: \[ \gamma = \frac{\mu}{L} \]

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The gyromagnetic ratio (\( \gamma \)) is a fundamental constant for a given particle or system that relates its magnetic properties to its rotational (mechanical) properties. It quantifies the magnetic moment generated by the angular momentum of the particle.
Step 2: Key Formula or Approach:
The definition of the gyromagnetic ratio for a particle is the ratio of its magnetic dipole moment (\( \vec{\mu} \)) to its angular momentum (\( \vec{L} \)).
\[ \gamma = \frac{|\vec{\mu}|}{|\vec{L}|} \] For an orbiting electron, the orbital magnetic moment is \( \mu_L \) and the orbital angular momentum is \( L \).
Step 3: Detailed Explanation:
According to the Bohr model for an electron revolving in an orbit, its motion constitutes a current loop. This current loop possesses a magnetic dipole moment. The electron also has orbital angular momentum due to its motion.
- The orbital magnetic moment is given by \( \mu_L = \frac{e}{2m_e} L \), where \( e \) is the elementary charge, \( m_e \) is the mass of the electron, and \( L \) is the magnitude of the orbital angular momentum.
- Rearranging this formula gives the ratio:
\[ \frac{\mu_L}{L} = \frac{e}{2m_e} \] This ratio, \( \frac{\mu_L}{L} \), is defined as the gyromagnetic ratio. Thus, it is the ratio of the magnetic moment to the angular momentum.
Let's check the options:
(A) and (D) are incorrect. The ratio of charge to mass (\(e/m\)) is the specific charge.
(B) and (E) are incorrect as the ratio involves angular momentum, not angular acceleration or angular velocity.
(C) correctly states the definition.
Step 4: Final Answer:
The gyromagnetic ratio of an electron is the ratio between its magnetic moment and its angular momentum. This corresponds to option (C).