In Figure below, an electron accelerated from rest through potential difference V1=0.955 kV enters the gap between two parallel plates having separation d = 18.2 mm and potential difference V2= 160 V. The lower plate is at the lower potential. Neglect fringing and assume that the electron's velocity vector is perpendicular to the electric field vector between the plates. In unit-vector notation, what uniform magnetic field allows the electron to travel in a straight line in the gap?
| | -----------------
| | |
| X | ----------> | d V2
| | |
V1 ----------------
Y
|
|
|_____ X
Dont you want the force on the electron from the 160V plates to equal the force from the magnetic field?
I will be happy to critique your thinking.
To find the uniform magnetic field that allows the electron to travel in a straight line in the gap between the plates, we need to analyze the forces acting on the electron.
In this scenario, the electric force on the electron is balanced by the magnetic force in order for the electron to move in a straight line.
1. Electric Force (Fe):
The electric force acting on the electron is given by the equation Fe = q * E, where q is the charge of the electron and E is the electric field between the plates.
2. Magnetic Force (Fm):
The magnetic force acting on the electron is given by the equation Fm = q * v * B, where q is the charge of the electron, v is the velocity of the electron, and B is the magnetic field.
As the electron is moving in a straight line, we know that the net force on the electron is zero. Therefore, we can equate the electric force and the magnetic force:
Fe = Fm
q * E = q * v * B
As the electron's velocity vector is perpendicular to the electric field vector, the electric field and the velocity are parallel to each other. Therefore, we can rewrite the equation as:
|E| = |v| * |B|
Now, we can substitute the given values into the equation to find the magnitude of the magnetic field |B|:
|B| = |E| / |v|
To find the unit vector notation for the magnetic field, we need to determine the direction. Given the geometry of the problem, the electric field is directed vertically upward, and the velocity vector of the electron is in the x-direction.
Therefore, the magnetic field should be directed in the negative y-direction (since the force on a negative charge moving in the positive x-direction experiences a magnetic force in the negative y-direction).
The unit-vector notation for the magnetic field is -j, where j is the unit vector in the y-direction.
Therefore, the answer in unit-vector notation is -|B| * j.