A particle of charge `q` and mass `m` moves in a circular orbit of radius `r` with angular speed `omega`. The ratio of the magnitude of its magnetic moment to that of its angular momentum depends on

Options

A. `omega` and `q`
B. `omega`, `q` and `m`
C. `q` and `m`
D. `omega` and `m`

Answer

To solve the problem step by step, we need to find the ratio of the magnetic moment (μ) to the angular momentum (L) of a charged particle moving in a circular orbit.

### Step 1: Define Angular Momentum (L)
The angular momentum \( L \) of a particle of mass \( m \) moving in a circular path of radius \( r \) with angular speed \( \omega \) is given by:
\[ L = I \omega \]
where \( I \) is the moment of inertia. For a point mass, the moment of inertia \( I \) is:
\[ I = m r^2 \]
Thus, we can write:
\[ L = m r^2 \omega \]

### Step 2: Define Magnetic Moment (μ)
The magnetic moment \( \mu \) of a charged particle moving in a circular orbit can be expressed as:
\[ \mu = I \cdot A \]
where \( A \) is the area of the circular orbit. The area \( A \) for a circle is given by:
\[ A = \pi r^2 \]
The current \( I \) due to the moving charge \( q \) is:
\[ I = \frac{q}{T} \]
where \( T \) is the time period of one complete revolution. The time period \( T \) can be expressed in terms of the angular speed \( \omega \):
\[ T = \frac{2\pi}{\omega} \]
Thus, the current becomes:
\[ I = \frac{q \omega}{2\pi} \]
Now substituting this into the expression for magnetic moment:
\[ \mu = \left(\frac{q \omega}{2\pi}\right) \cdot \pi r^2 = \frac{q \omega r^2}{2} \]

### Step 3: Find the Ratio of Magnetic Moment to Angular Momentum
Now we can find the ratio of the magnetic moment \( \mu \) to the angular momentum \( L \):
\[ \frac{\mu}{L} = \frac{\frac{q \omega r^2}{2}}{m r^2 \omega} \]
Simplifying this expression:
\[ \frac{\mu}{L} = \frac{q \omega r^2}{2 m r^2 \omega} = \frac{q}{2m} \]

### Conclusion
The ratio of the magnitude of the magnetic moment to that of the angular momentum is:
\[ \frac{\mu}{L} = \frac{q}{2m} \]
This shows that the ratio depends on the charge \( q \) and mass \( m \) of the particle.