In the previous problem, instead of being stationary, the system enters the magnetic field with a velocity `0.5 m// sec` along y-direction. Along which direction the system will move now?

Options

A. along `ahat(i)+bhat(j)`
B. along a circular path
C. system will become stationary
D. along a direction in x-y plane

Explanation

To solve the problem of the system entering a magnetic field with a velocity of 0.5 m/s along the y-direction, we will analyze the forces acting on the charged particle and the resulting motion.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Velocity of the system (v) = 0.5 m/s along the y-direction.
- Magnetic field (B) is directed out of the plane (which we can assume is in the z-direction).

2. **Set Up the Coordinate System:**
- Let the x-axis be horizontal, the y-axis be vertical, and the z-axis be out of the plane (towards the observer).
- The velocity vector can be represented as \( \vec{v} = 0 \hat{i} + 0.5 \hat{j} + 0 \hat{k} \).

3. **Determine the Magnetic Force:**
- The magnetic force \( \vec{F} \) on a charged particle moving in a magnetic field is given by the equation:
\[ \vec{F} = q (\vec{v} \times \vec{B}) \]
- Here, \( \vec{B} \) is directed out of the plane, which we can denote as \( \vec{B} = 0 \hat{i} + 0 \hat{j} + B \hat{k} \).

4. **Calculate the Cross Product:**
- To find the direction of the force, we calculate the cross product \( \vec{v} \times \vec{B} \):
\[ \vec{v} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0.5 & 0 \\ 0 & 0 & B \end{vmatrix} \]
- Expanding this determinant, we get:
\[ \vec{v} \times \vec{B} = \hat{i}(0.5 \cdot B) - \hat{j}(0) + \hat{k}(0) = 0.5B \hat{i} \]
- This indicates that the magnetic force \( \vec{F} \) is directed along the x-direction.

5. **Determine the Motion of the System:**
- Since the magnetic force acts perpendicular to the velocity of the particle, the particle will undergo circular motion in the plane perpendicular to the magnetic field.
- The radius of this circular motion will depend on the charge of the particle, its speed, and the strength of the magnetic field.

6. **Conclusion:**
- The system will move in a circular path due to the magnetic force acting on it, which is directed in the x-direction.

### Final Answer:
The system will move in a circular path.