(a) First, sketch a picture of the situation, showing the electron at its initial location and final location as it exits the region between the plates.
Figure: The initial position and momentum and final position and momentum of the electron
Now, draw a freebody diagram and add the forces on the electron to calculate the net force on the electron.
Figure: The freebody diagram for the electron.
This case is easy because there is only one force acting on the electron, and it's the electric force acting to the left. Thus,
Note that the change in momentum of the electron is in the direction of the net force on the electron, according to the momentum principle. The picture shows the direction of the change in momentum of the electron as it passes through the region between the plates.
Figure: The change in momentum of the electron.
Because there is no force in the y or z direction, then the ymomentum and zmomentum of the electron will be constant. They are:
and
However, the xmomentum changes according to the momentum principle. The initial xmomentum is zero ( ); therefore,
We cannot solve for the final xmomentum or xvelocity because we do not know the time interval. We must find the time (or clock reading) when the electron exits the region between the plates. We know that the ydisplacement of the electron as it travels between the plates is . Thus
Use the arithmetic mean for the average velocity. Note that the yvelocity does not change since there is no net force in the ydirection. As a result, the final yvelocity and initial yvelocity are the same, thus the average yvelocity is equal to the initial yvelocity.
Use the definition of average velocity to solve for the time interval.
Now, go back to the momentum principle and substitute the time interval to calculate the final velocity and final momentum of the electron.
To find out how far the electron is deflected to the left, use the definition of average velocity, for the xdirection.
The electron is deflected (which is 2.20 cm to the left). We now have found all of the quantities for the final position and final momentum of the electron. They are:
and
Compare the answers to the picture that we drew of the electron. As it exits the region between the plates, it has been deflected to the left and it has a momentum that is downward and to the left. This is in agreement with the directions we found for the final position and momentum of the electron.
