A particle of charge 'q' and mass 'm' move in a circular orbit of radius 'r' with frequency 'v' the ratio of the magnetic moment to angular momentum is:

Options

A. `(2qv)/(m)`
B. `(qv)/(2m)`
C. `(q)/(2mr)`
D. `(q)/(2m)`

Answer

To find the ratio of the magnetic moment to the angular momentum for a particle of charge 'q' and mass 'm' moving in a circular orbit of radius 'r' with frequency 'v', we can follow these steps:

### Step 1: Calculate the Magnetic Moment (M)
The magnetic moment (M) for a charged particle moving in a circular path is given by the formula:
\[ M = I \cdot A \]
where \( I \) is the current and \( A \) is the area of the circular path.

1. **Current (I)**: The current due to the moving charge is defined as the charge passing through a point per unit time. For a charge 'q' moving in a circular path, the current can be expressed as:
\[ I = \frac{q}{T} \]
where \( T \) is the time period of one complete revolution.

2. **Area (A)**: The area of the circular path is:
\[ A = \pi r^2 \]

Thus, the magnetic moment can be expressed as:
\[ M = \frac{q}{T} \cdot \pi r^2 \]

### Step 2: Express Time Period (T) in terms of Frequency (v)
The time period \( T \) is the reciprocal of the frequency \( v \):
\[ T = \frac{1}{v} \]

Substituting this into the equation for magnetic moment:
\[ M = qv \cdot \pi r^2 \]

### Step 3: Calculate the Angular Momentum (L)
The angular momentum (L) of the particle is given by:
\[ L = mvr \]
where \( v \) is the linear speed of the particle.

### Step 4: Find the Ratio of Magnetic Moment to Angular Momentum
Now, we need to find the ratio of the magnetic moment to the angular momentum:
\[ \text{Ratio} = \frac{M}{L} \]

Substituting the expressions for \( M \) and \( L \):
\[ \text{Ratio} = \frac{qv \cdot \pi r^2}{mvr} \]

### Step 5: Simplify the Ratio
In the ratio, we can cancel out \( v \) and \( r \):
\[ \text{Ratio} = \frac{q \cdot \pi r}{2m} \]

### Final Result
Thus, the ratio of the magnetic moment to the angular momentum is:
\[ \frac{M}{L} = \frac{q}{2m} \]