To determine in which case the charged particle will describe a helical path with the axis along the positive z-direction, we need to analyze the given scenarios based on the relationship between the velocity vector \(\vec{v}\), electric field \(\vec{E}\), and magnetic field \(\vec{B}\).
### Step 1: Understand the conditions for a helical path
A charged particle moves in a helical path when:
1. There is a component of velocity along the direction of the magnetic field.
2. The electric and magnetic forces must be perpendicular to each other.
3. The magnetic force provides the centripetal force necessary for circular motion, while the electric force provides the linear acceleration along the direction of the magnetic field.
### Step 2: Analyze each case
Let's analyze each case provided in the columns:
1. **Case (i)**:
- \(\vec{v} = \frac{2E_0}{B_0} \hat{x}\)
- \(\vec{E} = E_0 \hat{z}\)
- \(\vec{B} = -B_0 \hat{x}\)
Here, the velocity is in the x-direction, the electric field is in the z-direction, and the magnetic field is also in the x-direction. The magnetic force will not contribute to motion in the z-direction, so this case cannot describe a helical path along the z-axis.
2. **Case (ii)**:
- \(\vec{v} = \frac{E_0}{B_0} \hat{y}\)
- \(\vec{E} = -E_0 \hat{x}\)
- \(\vec{B} = B_0 \hat{x}\)
The velocity is in the y-direction, the electric field is in the negative x-direction, and the magnetic field is in the x-direction. The electric force acts in the negative x-direction while the magnetic force acts in the z-direction (due to the cross product). This case does not have a component of velocity in the z-direction, so it cannot describe a helical path along the z-axis.
3. **Case (iii)**:
- \(\vec{v} = 0\)
- \(\vec{E} = -E_0 \hat{x}\)
- \(\vec{B} = B_0 \hat{y}\)
Here, the particle is initially at rest. Since the velocity is zero, it cannot describe any path, let alone a helical path.
4. **Case (iv)**:
- \(\vec{v} = \frac{2E_0}{B_0} \hat{x}\)
- \(\vec{E} = E_0 \hat{x}\)
- \(\vec{B} = B_0 \hat{z}\)
In this case, the velocity is in the x-direction, the electric field is also in the x-direction, and the magnetic field is in the z-direction. The electric force will act in the x-direction, while the magnetic force will act in the y-direction (due to the cross product of velocity and magnetic field). This case does not have a component of velocity in the z-direction, so it cannot describe a helical path along the z-axis.
### Conclusion
None of the cases provided in the columns lead to a helical path along the positive z-direction. However, if we were to modify the conditions so that the velocity had a component in the z-direction while maintaining the perpendicularity of the forces, we could achieve a helical path.
