To solve this problem, we need to analyze the motion of an electron in the presence of both electric and magnetic fields. The electric field \( \mathbf{E} \) is directed along the negative Y-axis, while the magnetic field \( \mathbf{B} \) is directed along the negative Z-axis. The goal is to find the displacement of the electron along the Y-axis when its velocity becomes perpendicular to the electric field for the first time.
Understanding the Forces Acting on the Electron
When the electron is released, it experiences a force due to the electric field given by:
Electric Force: \( \mathbf{F_E} = q \mathbf{E} \)
Since the charge of the electron \( q \) is negative, the force will act in the direction opposite to the electric field. Therefore, if \( \mathbf{E} \) is in the negative Y-direction, the electric force \( \mathbf{F_E} \) will act in the positive Y-direction.
Magnetic Force and Motion
The magnetic force acting on the electron is given by:
Magnetic Force: \( \mathbf{F_B} = q (\mathbf{v} \times \mathbf{B}) \)
Here, \( \mathbf{v} \) is the velocity of the electron, and \( \mathbf{B} \) is the magnetic field. The direction of the magnetic force depends on the velocity of the electron and the direction of the magnetic field. As the electron moves, it will experience a magnetic force that is perpendicular to both its velocity and the magnetic field.
Equations of Motion
Initially, the electron starts from rest, so its initial velocity \( \mathbf{v_0} = 0 \). As it accelerates due to the electric field, its velocity will increase in the positive Y-direction. The acceleration \( \mathbf{a} \) due to the electric field can be expressed as:
Acceleration: \( \mathbf{a} = \frac{\mathbf{F_E}}{m} = \frac{q \mathbf{E}}{m} \)
Where \( m \) is the mass of the electron. The electron will continue to accelerate in the positive Y-direction until its velocity becomes perpendicular to the electric field.
Finding the Condition for Perpendicular Velocity
For the velocity to be perpendicular to the electric field, the velocity vector \( \mathbf{v} \) must have no component in the Y-direction. This means that the velocity will have to gain a component in the X-direction due to the magnetic force. The magnetic force will cause the electron to move in a circular path in the X-Y plane.
Calculating Displacement Along the Y-Axis
To find the displacement along the Y-axis when the velocity becomes perpendicular to the electric field, we can use the following steps:
- Calculate the time \( t \) it takes for the electron to reach a velocity \( v \) such that the magnetic force starts to influence its motion significantly.
- Use the relationship between the electric field and the resulting acceleration to find the velocity at that time.
- Determine the displacement in the Y-direction using the kinematic equation: \( y = v_0 t + \frac{1}{2} a t^2 \), where \( v_0 = 0 \).
Assuming the electric field \( E \) is constant, the displacement \( y \) can be calculated as:
Displacement: \( y = \frac{1}{2} \left(\frac{qE}{m}\right) t^2 \)
Final Thoughts
In summary, the displacement of the electron along the Y-axis when its velocity becomes perpendicular to the electric field can be determined by analyzing the forces acting on it and applying the equations of motion. The interplay between the electric and magnetic forces leads to a complex trajectory, but the fundamental principles of motion allow us to calculate the desired displacement effectively.
