A charged particle enters a unifrom magnetic field with velocity vector at an angle of `45^(@)` with the magnetic field . The pitch of the helical path followed by the particles is `rho` . The radius of the helix will be

Options

A. `(p)/(sqrt(2pi))`
B. `sqrt(2)P`
C. `(p)/(2pi)`
D. `sqrt(2p)/(pi)`

Answer

To solve the problem, we need to find the radius of the helical path followed by a charged particle moving in a uniform magnetic field at an angle of \(45^\circ\) to the field. We will use the relationships between the pitch of the helix and the radius.

### Step-by-Step Solution:

1. **Understanding the Motion**:
When a charged particle moves in a magnetic field, it follows a helical path if it enters the field at an angle. The motion can be decomposed into two components: one parallel to the magnetic field and the other perpendicular to it.

2. **Pitch of the Helical Path**:
The pitch \(p\) of the helical path is given by the formula:
\[ p = \frac{2\pi mv}{qB \cos \theta} \]
where:
- \(m\) is the mass of the particle,
- \(v\) is the speed of the particle,
- \(q\) is the charge of the particle,
- \(B\) is the magnetic field strength,
- \(\theta\) is the angle between the velocity vector and the magnetic field.

3. **Radius of the Helical Path**:
The radius \(r\) of the helical path is given by:
\[ r = \frac{mv \sin \theta}{qB} \]

4. **Dividing the Two Equations**:
To find the relationship between the pitch \(p\) and the radius \(r\), we can divide the two equations:
\[ \frac{p}{r} = \frac{\frac{2\pi mv}{qB \cos \theta}}{\frac{mv \sin \theta}{qB}} = \frac{2\pi \cos \theta}{\sin \theta} \]
This simplifies to:
\[ \frac{p}{r} = 2\pi \cot \theta \]

5. **Finding the Radius in Terms of Pitch**:
Rearranging the equation gives:
\[ r = \frac{p}{2\pi \cot \theta} \]

6. **Substituting the Angle**:
Since the angle \(\theta\) is given as \(45^\circ\), we know that:
\[ \cot 45^\circ = 1 \]
Therefore, substituting this value into the equation for \(r\):
\[ r = \frac{p}{2\pi \cdot 1} = \frac{p}{2\pi} \]

### Final Answer:
The radius of the helix is:
\[ r = \frac{p}{2\pi} \]