Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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1250002     Momentum and position for an electron moving between charged plates with a constant electric force on the electron.     1250002
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Question

 

An electron of mass and initial speed in the –y direction enters a region between charged plates in which the force on the electron is constant, in the –x direction with a magnitude . After the electron travels 5 cm in the -y direction, it exits the region between the plates. At this instant when it exits, what is its momentum and how far has it deflected to the left from where it started?


Figure: An electron travels between two charged plates.

 



Solution

 

(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 free-body diagram and add the forces on the electron to calculate the net force on the electron.


Figure: The free-body 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 y-momentum and z-momentum of the electron will be constant. They are:

and

However, the x-momentum changes according to the momentum principle. The initial x-momentum is zero ( ); therefore,

We cannot solve for the final x-momentum or x-velocity 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 y-displacement of the electron as it travels between the plates is . Thus

Use the arithmetic mean for the average velocity. Note that the y-velocity does not change since there is no net force in the y-direction. As a result, the final y-velocity and initial y-velocity are the same, thus the average y-velocity is equal to the initial y-velocity.

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 x-direction.

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.