In a cyclotron, the angular frequency of a charged particle is independent

Options

A. mass
B. speed
C. charge
D. magnetic field

Explanation

To solve the question regarding the angular frequency of a charged particle in a cyclotron, we need to analyze the relationship between angular frequency and other quantities involved in the cyclotron's operation.

### Step-by-Step Solution:

1. **Understanding Angular Frequency**:
Angular frequency (ω) is defined as the rate of change of the angle with respect to time. It can be expressed mathematically as:
\[ \omega = \frac{2\pi}{T} \]
where \(T\) is the time period of the motion.

2. **Time Period in a Cyclotron**:
In a cyclotron, the time period \(T\) for a charged particle moving in a magnetic field is given by:
\[ T = \frac{2\pi M}{QB} \]
where:
- \(M\) is the mass of the particle,
- \(Q\) is the charge of the particle,
- \(B\) is the magnetic field strength.

3. **Substituting Time Period into Angular Frequency**:
We can substitute the expression for \(T\) into the equation for angular frequency:
\[ \omega = \frac{2\pi}{T} = \frac{2\pi}{\frac{2\pi M}{QB}} = \frac{QB}{M} \]

4. **Analyzing the Relationship**:
From the equation \(\omega = \frac{QB}{M}\), we can see that:
- Angular frequency \(\omega\) is directly proportional to the charge \(Q\) (i.e., \(\omega \propto Q\)).
- Angular frequency \(\omega\) is directly proportional to the magnetic field \(B\) (i.e., \(\omega \propto B\)).
- Angular frequency \(\omega\) is inversely proportional to the mass \(M\) (i.e., \(\omega \propto \frac{1}{M}\)).

5. **Identifying the Independent Quantity**:
The only quantity that does not appear in the expression for angular frequency is the speed (or velocity) of the particle. Therefore, the angular frequency \(\omega\) is independent of the speed of the charged particle.

6. **Conclusion**:
Based on the analysis, we conclude that the angular frequency of a charged particle in a cyclotron is independent of its speed. Thus, the answer to the question is:
**Speed (Option B)**.