A charged particle is moving in a cyclotron, what effect on the radius of path of this charged particle will occur when the frequency of the radio frequency field is doubled?

Options

A. It will also be doubled.
B. It will be halved.
C. It will be increased by four times.
D. It will remain unchanged.

Explanation

To solve the problem, we need to analyze the relationship between the radius of the path of a charged particle in a cyclotron and the frequency of the radio frequency field.

### Step-by-Step Solution:

1. **Understanding the Cyclotron Motion**:
- A charged particle moving in a magnetic field experiences a centripetal force that keeps it in circular motion. This force is provided by the magnetic force acting on the charged particle.

2. **Equating Forces**:
- The centripetal force acting on the particle can be expressed as:
\[ F_c = \frac{mv^2}{r} \]
- The magnetic force acting on the charged particle is given by:
\[ F_m = qvB \]
- Setting these two forces equal gives:
\[ \frac{mv^2}{r} = qvB \]

3. **Solving for Radius (R)**:
- Rearranging the equation to solve for the radius \( r \):
\[ r = \frac{mv}{qB} \]
- This equation shows that the radius \( r \) depends on the mass \( m \), velocity \( v \), charge \( q \), and magnetic field \( B \).

4. **Understanding Frequency**:
- The frequency \( f \) of the charged particle's motion in the cyclotron can be derived from the time period \( T \):
\[ T = \frac{2\pi m}{qB} \]
- The frequency is the reciprocal of the time period:
\[ f = \frac{1}{T} = \frac{qB}{2\pi m} \]

5. **Effect of Doubling Frequency**:
- If the frequency \( f \) is doubled, it implies:
\[ f' = 2f = \frac{qB'}{2\pi m} \]
- This means that the magnetic field \( B \) must also change in order to maintain the relationship, but the radius \( r \) does not depend on frequency directly.

6. **Conclusion**:
- Since the radius \( r \) is independent of the frequency \( f \), doubling the frequency of the radio frequency field does not change the radius of the path of the charged particle in the cyclotron. Therefore, the radius remains unchanged.

### Final Answer:
The radius of the path of the charged particle will remain unchanged when the frequency of the radio frequency field is doubled.

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