Electron Gun Equation Question

[rorygolledge]

I'm having trouble understanding how the electron gun equation can predict final velocity of an electron when accelerated between plates when no "distance travelled" variable is included,

"In an electron gun, an electron is accelerated by a potential difference of 28 kV.

At what speed will the electrons exit the assembly?"

I understand to derive the electron gun equation, we let u = 0 and change in kinetic energy to equal work done.

(1/2)*mv^2 = qV

so v = sqrt((2qv)/m)

My problem is that shouldn't the final velocity given be different based on how long the electron is in the field for?

It just seems to fall apart for me as I see that if it is only in the electric field for 1cm, it has the same final velocity when exiting as if it were in the field for 100m when using this equation.

[DrDusk]

I'm having trouble understanding how the electron gun equation can predict final velocity of an electron when accelerated between plates when no "distance travelled" variable is included,

"In an electron gun, an electron is accelerated by a potential difference of 28 kV.

At what speed will the electrons exit the assembly?"

I understand to derive the question, we let u = 0 and change in kinetic energy to equal work done.

(1/2)*mv^2 = qV

so v = sqrt((2qv)/m)

My problem is that shouldn't the final velocity given be different based on how long the electron is in the field for?

It just seems to fall apart for me as I see that if it is only in the electric field for 1cm, it has the same final velocity when exiting as if it were in the field for 100m when using this equation.

You are right, and here's why.

Let's say the electron travels a distance d_1 in the electron gun, and the distance between the plates is D. Now let's use the fact that work is the change in Kinetic energy and we get



However now let's say the electron is inside the plates for a time that's less than the time it takes to travel the full length of the plates. This means we CANNOT say d_1 = D. Which leaves us with



Which we can see depends on the distance d_1 that the electron travels between the plates, and since distance depends on time and acceleration, it shows that the velocity DOES depend on the amount of time the electron spends between the plates. We just cancel the two 'd's' because usually we say the electron travels the full distance between the plates.

Good question! =)

[rorygolledge]

You are right, and here's why.

Let's say the electron travels a distance d_1 in the electron gun, and the distance between the plates is D. Now let's use the fact that work is the change in Kinetic energy and we get



However now let's say the electron is inside the plates for a time that's less than the time it takes to travel the full length of the plates. This means we CANNOT say d_1 = D. Which leaves us with



Which we can see depends on the distance d_1 that the electron travels between the plates, and since distance depends on time and acceleration, it shows that the velocity DOES depend on the amount of time the electron spends between the plates. We just cancel the two 'd's' because usually we say the electron travels the full distance between the plates.

Good question! =)

Thank you, that makes a lot more sense now, had been bugging me for a while haha.