In the previous question, if the pitch of the helical path is equal to the maximum distance of the particle from the x-axis

Options

A. `cos theta=1/pi`
B. `sin theta=1/pi`
C. `tan theta=1/pi`
D. `tan theta=pi`

Explanation

To solve the problem step by step, we need to analyze the relationship between the pitch of the helical path and the maximum distance of the particle from the x-axis.

### Step 1: Understand the definitions
- **Pitch of the helical path**: The vertical distance covered in one complete turn of the helix. It is given by the formula:
\[ \text{Pitch} = \frac{2\pi m}{qBv} \cos \theta \]
where \( m \) is mass, \( q \) is charge, \( B \) is the magnetic field strength, \( v \) is the velocity of the particle, and \( \theta \) is the angle of the velocity vector with respect to the magnetic field.

- **Maximum distance from the x-axis**: This is equal to the diameter of the circular motion in the xy-plane, which is given by:
\[ \text{Maximum distance} = 2r \]
where \( r \) is the radius of the circular motion, and it is given by:
\[ r = \frac{mv \sin \theta}{qB} \]

### Step 2: Set up the equation
Since we know that the pitch is equal to the maximum distance from the x-axis, we can write:
\[ \frac{2\pi m}{qBv} \cos \theta = 2 \left( \frac{mv \sin \theta}{qB} \right) \]

### Step 3: Simplify the equation
1. Cancel out common terms on both sides:
- The \( 2 \) cancels out.
- The \( m \) cancels out.
- The \( qB \) cancels out.

This gives us:
\[ \pi \cos \theta = v \sin \theta \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ \tan \theta = \frac{\pi}{v} \]

### Step 5: Analyze the relationship
From the equation \( \tan \theta = \frac{\pi}{v} \), we can derive that if we set \( v = \pi \), then:
\[ \tan \theta = \pi \]

### Conclusion
Thus, we have shown that if the pitch of the helical path is equal to the maximum distance of the particle from the x-axis, then:
\[ \tan \theta = \pi \]