When a positively charged particle enters a uniform magnetic field with uniform velocity, its trajectory can be
`a)` a straight line `b)` a circle `c)` a helix

Options

A. (i) only
B. (i) or (ii)
C. (i) or (iii)
D. any one of (i),(ii) and (iii)

Answer

To solve the question regarding the trajectory of a positively charged particle entering a uniform magnetic field with uniform velocity, we will analyze the different scenarios based on the angle between the velocity of the particle and the magnetic field.

### Step-by-Step Solution:

1. **Understanding the Force on a Charged Particle in a Magnetic Field**:
The force experienced by a charged particle moving in a magnetic field is given by the equation:
\[ F = q \mathbf{v} \times \mathbf{B} \]
where \( F \) is the magnetic force, \( q \) is the charge of the particle, \( \mathbf{v} \) is the velocity vector, and \( \mathbf{B} \) is the magnetic field vector.

2. **Analyzing the Angle Between Velocity and Magnetic Field**:
The angle \( \theta \) between the velocity of the charged particle and the magnetic field affects the force:
\[ F = qvB \sin \theta \]
- If \( \theta = 0^\circ \) or \( \theta = 180^\circ \): The particle is moving parallel or anti-parallel to the magnetic field. In this case, \( \sin \theta = 0 \), so the magnetic force \( F = 0 \). Therefore, the trajectory will be a **straight line**.

3. **Case of \( \theta = 90^\circ \)**:
- If \( \theta = 90^\circ\): The particle moves perpendicular to the magnetic field. Here, \( \sin 90^\circ = 1 \), and the particle experiences a maximum magnetic force. The force acts as a centripetal force, causing the particle to move in a **circular path**.

4. **Case of \( 0^\circ < \theta < 90^\circ \) or \( 90^\circ < \theta < 180^\circ \)**:
- If the angle \( \theta \) is neither 0 nor 90 degrees, the particle has both a component of velocity parallel to the magnetic field and a component perpendicular to it. The perpendicular component will cause circular motion while the parallel component will cause linear motion along the direction of the magnetic field. This results in a **helical trajectory**.

5. **Conclusion**:
Based on the analysis:
- The trajectory can be a straight line (when \( \theta = 0^\circ \) or \( 180^\circ \)),
- A circle (when \( \theta = 90^\circ \)),
- A helix (when \( 0^\circ < \theta < 90^\circ \) or \( 90^\circ < \theta < 180^\circ \)).

Thus, the correct answer is that the trajectory can be **any of the options: a straight line, a circle, or a helix**.