The magnetic field exists along negative x-axis and electric field exists along positive x-axis. A charged particle moves with a velocity inclined at an angle `(theta)` with vertical in (x-y) plane. What will be the correct diagram of the helical path?

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the motion of a charged particle in the presence of both electric and magnetic fields. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Fields
- The magnetic field (B) is directed along the negative x-axis.
- The electric field (E) is directed along the positive x-axis.

### Step 2: Analyze the Velocity of the Charged Particle
- The charged particle moves with a velocity (V) that is inclined at an angle (θ) with the vertical in the x-y plane.
- We can resolve this velocity into two components:
- **Vertical Component (V_y)**: This is the component of velocity in the y-direction, given by \( V_y = V \cos(\theta) \).
- **Horizontal Component (V_x)**: This is the component of velocity in the x-direction, given by \( V_x = V \sin(\theta) \).

### Step 3: Determine the Forces Acting on the Particle
- The particle experiences a magnetic force due to the magnetic field. The magnetic force (F_m) is given by:
\[ F_m = q(V_y \times B) \]
where \( q \) is the charge of the particle and \( B \) is the magnetic field strength. Since the magnetic field is along the negative x-axis, the force will act in a direction perpendicular to both the velocity and the magnetic field.

- The particle also experiences an electric force (F_e) due to the electric field, which is given by:
\[ F_e = qE \]
This force acts in the positive x-direction.

### Step 4: Motion of the Particle
- The component of velocity perpendicular to the magnetic field (V_y) will cause the particle to move in a circular path due to the magnetic force.
- The electric field will exert a constant force in the positive x-direction, resulting in an acceleration in that direction.

### Step 5: Resulting Path of the Particle
- The combination of circular motion due to the magnetic field and linear acceleration due to the electric field results in a helical path.
- As the particle accelerates in the x-direction, the radius of the circular motion remains constant, but the pitch of the helix increases.

### Step 6: Identify the Correct Diagram
- The correct diagram should show a helical path that is increasing in the x-direction, indicating that the particle is accelerating.
- Among the options provided, the one that shows the particle moving in a helical path with increasing x-coordinates (x1 < x2 < x3) is the correct answer.

### Conclusion
- The correct option is **Option D**, which illustrates the particle accelerating in the positive x-direction while moving in a helical path.