A charged particle enters into a magnetic field with velocity vector making an angle of `30^@` with respect of the direction of magnetic field. The path of the particle is

Options

A. circular
B. helical
C. elliptical
D. straight line

Explanation

To determine the path of a charged particle entering a magnetic field at an angle, we can follow these steps:

### Step 1: Understand the Setup
A charged particle (let's assume it has a charge \( q \)) enters a uniform magnetic field \( \mathbf{B} \) with a velocity \( \mathbf{V} \) that makes an angle of \( 30^\circ \) with the direction of the magnetic field.

**Hint:** Visualize the scenario by drawing the velocity vector and the magnetic field vector.

### Step 2: Break Down the Velocity Vector
The velocity vector \( \mathbf{V} \) can be resolved into two components:
- A component parallel to the magnetic field: \( V_{\parallel} = V \cos(30^\circ) \)
- A component perpendicular to the magnetic field: \( V_{\perpendicular} = V \sin(30^\circ) \)

**Hint:** Use trigonometric functions to resolve the velocity into components based on the angle given.

### Step 3: Analyze the Forces Acting on the Particle
The magnetic force acting on the charged particle is given by the Lorentz force equation:
\[ \mathbf{F} = q (\mathbf{V} \times \mathbf{B}) \]
The force is dependent on the angle between the velocity vector and the magnetic field vector.

**Hint:** Remember that the sine of the angle determines the effective component of the force acting perpendicular to the magnetic field.

### Step 4: Determine the Motion in Each Direction
1. **Parallel Component:** The component \( V_{\parallel} \) does not experience any magnetic force because it is parallel to the magnetic field. Thus, it continues to move in a straight line.

2. **Perpendicular Component:** The component \( V_{\perpendicular} \) experiences a magnetic force, which causes it to move in a circular path. The radius of this circular motion can be calculated using the formula:
\[ r = \frac{mv_{\perpendicular}}{qB} \]

**Hint:** Recall that a charged particle in a magnetic field experiences circular motion due to the perpendicular component of its velocity.

### Step 5: Combine the Motions
The combination of the straight-line motion (due to the parallel component) and the circular motion (due to the perpendicular component) results in a helical path. The particle spirals around the direction of the magnetic field while also moving forward.

**Hint:** Think of how a screw moves; it goes forward while rotating, similar to how the charged particle moves in this scenario.

### Conclusion
Thus, the path of the charged particle entering the magnetic field at an angle of \( 30^\circ \) is **helical**.

**Final Answer:** The path of the particle is helical.