The cyclotron frequency `v_(c)` is given by

Options

A. `(qB)/(2pim)`
B. `(mB)/(2piq)`
C. `(2pim)/(qB)`
D. `(2piB)/(qm)`

Explanation

To find the cyclotron frequency \( v_c \) of a charged particle moving in a circular path in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Forces Involved**: In a cyclotron, a charged particle moves in a circular path due to the magnetic field. The centripetal force required to keep the particle in circular motion is provided by the magnetic force acting on it.

2. **Write the Expression for Centripetal Force**: The centripetal force \( F_c \) acting on a particle of mass \( m \) moving with velocity \( v \) in a circular path of radius \( R \) is given by:
\[ F_c = \frac{mv^2}{R} \]

3. **Write the Expression for Magnetic Force**: The magnetic force \( F_B \) acting on a charged particle with charge \( Q \) moving with velocity \( v \) in a magnetic field \( B \) is given by:
\[ F_B = QvB \]

4. **Set the Forces Equal**: Since the centripetal force is provided by the magnetic force, we can set these two expressions equal to each other:
\[ \frac{mv^2}{R} = QvB \]

5. **Rearrange the Equation**: We can simplify this equation by dividing both sides by \( v \) (assuming \( v \neq 0 \)):
\[ \frac{mv}{R} = QB \]

6. **Solve for Velocity**: Rearranging gives us the expression for velocity:
\[ v = \frac{QBR}{m} \]

7. **Relate Velocity to Frequency**: The relationship between the velocity \( v \) and the frequency \( v_c \) of the circular motion is given by:
\[ v = 2 \pi R v_c \]

8. **Substitute for Velocity**: Now, substituting the expression for \( v \) from step 6 into the frequency equation:
\[ \frac{QBR}{m} = 2 \pi R v_c \]

9. **Cancel \( R \)**: Since \( R \) appears on both sides, we can cancel it out (assuming \( R \neq 0 \)):
\[ \frac{QB}{m} = 2 \pi v_c \]

10. **Solve for Cyclotron Frequency**: Finally, we can solve for the cyclotron frequency \( v_c \):
\[ v_c = \frac{QB}{2 \pi m} \]

### Final Answer:
The cyclotron frequency \( v_c \) is given by:
\[ v_c = \frac{QB}{2 \pi m} \]