To determine how the speed \( v \) of a charged particle moving through a uniform magnetic field in the x-direction and a uniform electric field in the y-direction depends on its position, we need to analyze the forces acting on the particle and the nature of the fields involved.
Understanding the Forces at Play
When a charged particle moves through electric and magnetic fields, it experiences forces due to both fields. The electric force (\( \mathbf{F}_E \)) acting on the particle is given by:
Electric Force: \( \mathbf{F}_E = q \mathbf{E} \)
where \( q \) is the charge of the particle and \( \mathbf{E} \) is the electric field vector. In this case, since the electric field is uniform and directed in the y-direction, the force will also be constant in that direction.
The magnetic force (\( \mathbf{F}_B \)) acting on the particle is given by:
Magnetic Force: \( \mathbf{F}_B = q (\mathbf{v} \times \mathbf{B}) \)
Here, \( \mathbf{B} \) is the magnetic field vector, and \( \mathbf{v} \) is the velocity of the particle. The magnetic force depends on the velocity of the particle and is perpendicular to both the velocity and the magnetic field direction.
Analyzing the Motion
As the particle moves, it will experience a continuous change in velocity due to the forces acting on it. The electric field will accelerate the particle in the y-direction, while the magnetic field will exert a force that depends on the particle's velocity. The key point is that the magnetic force does not do work on the particle; it only changes the direction of the velocity, not its magnitude.
Dependence on Position
Now, let’s consider how the speed \( v \) of the particle depends on its position:
- Only on x: This would imply that the speed changes solely based on the x-coordinate, which is not the case here since the electric field affects the y-direction.
- Only on y: Similar to the previous point, this would suggest that the speed is influenced only by the y-coordinate, which is also incorrect due to the magnetic field's influence.
- On both x and y, but not z: While the speed does depend on the electric field in the y-direction, the magnetic field's influence is also tied to the velocity in the x-direction, making this option incomplete.
- On x, y, and z: This option considers all three dimensions. The particle's speed is influenced by its position in the x and y directions due to the electric and magnetic fields, while the z-coordinate does not directly affect the forces but can influence the overall motion in a three-dimensional space.
Conclusion on Speed Dependence
Based on the analysis, the correct conclusion is that the speed \( v \) of the charged particle depends on its position in all three dimensions: x, y, and z. The electric field accelerates the particle in the y-direction, while the magnetic field affects the particle's trajectory based on its velocity in the x-direction. Therefore, the answer is that \( v \) depends on x, y, and z.
