There is a uniform magnetic field and a charged particle is given an initial velocity at an acute angle to the direction of the magnetic field. What kind of path will the particle follow?

Options

A. Circle
B. Straight line
C. Helical path with a non-uniform pitch.
D. Helical path with a uniform pitch.

Answer

To determine the path of a charged particle moving in a uniform magnetic field at an acute angle to the field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Situation**:
- We have a uniform magnetic field (B) and a charged particle (with charge Q) that is given an initial velocity (V) at an acute angle (θ) to the direction of the magnetic field.

2. **Break Down the Velocity**:
- The initial velocity can be resolved into two components:
- **Parallel to the magnetic field**: \( V_{x} = V \cos \theta \)
- **Perpendicular to the magnetic field**: \( V_{y} = V \sin \theta \)

3. **Analyze the Forces**:
- The magnetic force acting on the charged particle is given by:
\[ F_{B} = Q V_{y} B = Q (V \sin \theta) B \]
- This force acts perpendicular to both the velocity of the particle and the magnetic field, causing the particle to experience circular motion in the plane perpendicular to the magnetic field.

4. **Determine the Circular Motion**:
- The component of velocity \( V_{y} \) (which is \( V \sin \theta \)) causes the particle to move in a circular path. The radius of this circular motion can be derived from the centripetal force equation:
\[ F_{B} = \frac{m V_{y}^2}{R} \]
- Setting the magnetic force equal to the centripetal force gives:
\[ Q (V \sin \theta) B = \frac{m (V \sin \theta)^2}{R} \]
- Rearranging this equation allows us to find the radius \( R \):
\[ R = \frac{m V \sin \theta}{Q B} \]

5. **Consider the Motion Along the Magnetic Field**:
- The component of velocity \( V_{x} = V \cos \theta \) does not experience any magnetic force, meaning it remains constant. This results in the particle moving linearly along the direction of the magnetic field.

6. **Combine the Motions**:
- The combination of circular motion (due to \( V_{y} \)) and linear motion (due to \( V_{x} \)) results in a helical path. The pitch of the helix is determined by the linear motion along the magnetic field.

7. **Conclusion**:
- The charged particle will follow a helical path with a uniform pitch, as it moves in a circle while simultaneously advancing along the direction of the magnetic field.

### Final Answer:
The charged particle will follow a **helical path with a uniform pitch**.

---