In the cyclotron, as radius of the circular path of the charged particle increase (`omega` = angular velocity, v = linear velocity)

Options

A. Both `omega and upsilon` increase
B. only `omega` increase, `upsilon` remains constant
C. only `upsilon` increases , `omega` remains constant
D. `upsilon` increases, `omega` decreases

Answer

To solve the problem regarding the behavior of angular velocity and linear velocity in a cyclotron as the radius of the circular path of a charged particle increases, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Cyclotron**:
A cyclotron is a device that accelerates charged particles using a magnetic field. The charged particle moves in a circular path due to the Lorentz force exerted by the magnetic field.

2. **Identify the Key Relationships**:
- The linear velocity \( v \) of the charged particle in a cyclotron is given by the formula:
\[ v = \frac{Bqr}{m} \]
where:
- \( B \) = magnetic field strength
- \( q \) = charge of the particle
- \( r \) = radius of the circular path
- \( m \) = mass of the particle

- The angular velocity \( \omega \) is given by:
\[ \omega = \frac{v}{r} \]

3. **Analyzing the Effect of Increasing Radius**:
- If the radius \( r \) increases, we can analyze how it affects \( v \) and \( \omega \).
- From the equation for linear velocity:
\[ v = \frac{Bqr}{m} \]
Since \( B \), \( q \), and \( m \) are constants, an increase in \( r \) will lead to a proportional increase in \( v \). Thus, \( v \) increases as \( r \) increases.

4. **Analyzing Angular Velocity**:
- Now, substituting the expression for \( v \) into the equation for \( \omega \):
\[ \omega = \frac{v}{r} = \frac{Bqr/m}{r} = \frac{Bq}{m} \]
- Here, \( B \), \( q \), and \( m \) are constants, meaning that \( \omega \) does not depend on \( r \). Therefore, \( \omega \) remains constant even as \( r \) increases.

5. **Conclusion**:
- As the radius \( r \) of the circular path increases, the linear velocity \( v \) increases, while the angular velocity \( \omega \) remains constant.

### Final Answer:
- Linear velocity \( v \) increases.
- Angular velocity \( \omega \) remains constant.