In cyclotron for a given magnet radius of the semicircle traced by positive ion is directly proportional to (where v= velocity of positive ion)

Options

A. `v^(-2)`
B. `v^(-1)`
C. `v`
D. `v^(2)`

Answer

To solve the problem, we need to analyze the relationship between the radius of the semicircle traced by a positive ion in a cyclotron and its velocity.

### Step-by-Step Solution:

1. **Understanding the Cyclotron Motion**:
In a cyclotron, charged particles (like positive ions) move in circular paths due to the magnetic field. The motion is uniform circular motion, which means there is a centripetal force acting on the particle.

**Hint**: Recall that uniform circular motion requires a centripetal force directed towards the center of the circular path.

2. **Centripetal Force and Magnetic Force**:
The centripetal force (\(F_c\)) required to keep the particle moving in a circle is given by the formula:
\[ F_c = \frac{mv^2}{R} \]
where \(m\) is the mass of the ion, \(v\) is its velocity, and \(R\) is the radius of the circular path.

**Hint**: Remember that the centripetal force depends on the mass, velocity, and radius of the circular path.

3. **Magnetic Force**:
The magnetic force (\(F_m\)) acting on the charged particle moving in a magnetic field is given by:
\[ F_m = qvB \]
where \(q\) is the charge of the ion, \(v\) is its velocity, and \(B\) is the magnetic field strength.

**Hint**: The magnetic force depends on the charge, velocity, and strength of the magnetic field.

4. **Setting the Forces Equal**:
For the charged particle to move in a circular path, the centripetal force must equal the magnetic force:
\[ \frac{mv^2}{R} = qvB \]

**Hint**: Equate the expressions for centripetal force and magnetic force to find the relationship between \(R\) and \(v\).

5. **Rearranging the Equation**:
Rearranging the equation gives:
\[ R = \frac{mv}{qB} \]

**Hint**: Isolate \(R\) to see how it relates to \(v\).

6. **Analyzing the Relationship**:
From the equation \(R = \frac{mv}{qB}\), we see that for a given magnetic field (\(B\)), mass (\(m\)), and charge (\(q\)), the radius \(R\) is directly proportional to the velocity \(v\):
\[ R \propto v \]

**Hint**: Direct proportionality means that if one quantity increases, the other does as well.

7. **Conclusion**:
Therefore, the radius \(R\) of the semicircle traced by the positive ion in a cyclotron is directly proportional to the velocity \(v\) of the ion.

The correct answer is:
\[ R \propto v^{1} \]

Thus, the answer corresponds to option **C**.