An electron is initially at rest in, but not on the axis of, a solenoid that is carrying a steady current I directed as shown in the cross section below. The electron is initially at rest on the -x axis. The current in the solenoid is suddenly increaed to 2I. In what direction is the velocity vector of the electron immediately after this change?
To determine the direction of the velocity vector of the electron immediately after the current in the solenoid is increased, we can use the concept of the right-hand rule for electromagnetic fields.
1. Identify the direction of the current in the solenoid: The current is directed as shown in the cross section. Let's assume it is flowing from left to right.
2. Apply the right-hand rule: If you point the thumb of your right hand along the direction of the current in the solenoid, the fingers will naturally wrap around in the direction of the magnetic field lines. From the perspective of the electron, this would be a counterclockwise rotation of the fingers.
3. Determine the interaction between the magnetic field and the electron: Due to the magnetic field created by the current in the solenoid, there will be a force acting on the electron perpendicular to both its velocity and the magnetic field direction. This is known as the Lorentz force.
4. Apply the right-hand rule for the Lorentz force: To find the direction of the force on the electron, point your right-hand thumb in the direction of the electron's velocity, which is initially along the -x axis. Then, rotate your fingers in the counterclockwise direction of the magnetic field (due to the increased current in the solenoid).
5. Determine the resulting velocity direction: Since the force on the electron will be in a direction perpendicular to its initial velocity, it will acquire a velocity component perpendicular to the -x axis. Applying the right-hand rule, the resulting velocity vector of the electron will be directed out of the screen or into the page.
Therefore, the direction of the velocity vector of the electron immediately after the increase in the current in the solenoid will be perpendicular to the -x axis and either out of the screen or into the page, depending on the specific setup of the cross-section.