# Electron speed and acceleration in an electric field

• Antoine DEBRAY
In summary, the conversation revolves around the behavior of electrons in a constant and linear electric field. The speaker mentions being satisfied with equations and a graph that describe the speed of electrons based on the acceleration voltage, but later raises concerns about the lack of a speed transition between the electron source and any given place in the electric field. They question whether there may be a relativistic effect at play and express a desire to better understand this concept.f

#### Antoine DEBRAY

Hi everyone,

I often work on a SEM, a type of microscope which is based on electron acceleration between an electron source and the sample you are working on. For this reason and since a few weeks I was wondering how an electron (in term of speed) behaves in a constant and linear electric field and I have succeded to find some answers.

I was completely satisfied with several equations (source 1, source 2 and source 3) and with an article including a graph that matches results for every mentioned equations (source 4) due to the fact that they are all completely equivalent between each others. They properly describe the speed an electron can reach depending on the acceleration voltage (the voltage between the two sides of the electric field) and, due to the fact they take relativistic effect in account, doesn't allow eletron speed to be higher than light speed in the vacuum.

But something hit me a few days ago: following these equations, the electron speed is constant whatever the electron location is between the two sides of the electric field, like if there is absolutely no speed transition between the moment where the electron speed is roughly 0 (inside the electron source) and any given place inside the electric field.
Of course I am not comfortable with this idea because I guess that the speed that could be calculated is an asymptotic value, involving an acceleration phase.

If we use very basic equations (F=ma=eE [Force, mass, acceleration, electric charge and electric field value]) then we quickly see that acceleration should be constant with any given electric field, meaning that speed should contiously increase whitout any limit and that seems to be clearly in contradiction with electrons speed equations mentionned before.

I highly presume that there is, again, some kind of relativistic effect involved but I am really not sure and even if this is the case this is beyond my understanding of physics.
Even though I don't need to know this kind of details for my job, it bothers me more and more.

I thank you in advance for any suggestions. I don't understand the premise of your question, "source 1" that you linked says:
An electron flies into a homogenous electric field against the direction of the line of force. The initial electron velocity/speed is 10 km s-1. The electric field intensity is 20 V m-1. Calculate the speed/velocity of the electron when is has already moved/traveled 9 cm in the field.

So very clearly, the speed is not constant. I has an initial speed and it accelerates. The problem asks you to calculate the speed at a later time. I don't understand the premise of your question, "source 1" that you linked says:

So very clearly, the speed is not constant. I has an initial speed and it accelerates. The problem asks you to calculate the speed at a later time.

You are right, I was so concentrated on the solutions that I had forgotten that the first and second links start as numerical exercices but what has interested me in them are the solving steps and the solution. The given values in these exercices don't really matter and don't match the case I mentionned (electron microscopy) anyway.

By the way thank you to whoever have moved this thread in the right section, I had the feeling that "High energy physics" was not the perfect place but I haven't thought to "General Physics".

Hi everyone,

I often work on a SEM, a type of microscope which is based on electron acceleration between an electron source and the sample you are working on. For this reason and since a few weeks I was wondering how an electron (in term of speed) behaves in a constant and linear electric field and I have succeded to find some answers.

I was completely satisfied with several equations (source 1, source 2 and source 3) and with an article including a graph that matches results for every mentioned equations (source 4) due to the fact that they are all completely equivalent between each others. They properly describe the speed an electron can reach depending on the acceleration voltage (the voltage between the two sides of the electric field) and, due to the fact they take relativistic effect in account, doesn't allow eletron speed to be higher than light speed in the vacuum.

But something hit me a few days ago: following these equations, the electron speed is constant whatever the electron location is between the two sides of the electric field, like if there is absolutely no speed transition between the moment where the electron speed is roughly 0 (inside the electron source) and any given place inside the electric field.
Of course I am not comfortable with this idea because I guess that the speed that could be calculated is an asymptotic value, involving an acceleration phase.

If we use very basic equations (F=ma=eE [Force, mass, acceleration, electric charge and electric field value]) then we quickly see that acceleration should be constant with any given electric field, meaning that speed should contiously increase whitout any limit and that seems to be clearly in contradiction with electrons speed equations mentionned before.

I highly presume that there is, again, some kind of relativistic effect involved but I am really not sure and even if this is the case this is beyond my understanding of physics.
Even though I don't need to know this kind of details for my job, it bothers me more and more.

I thank you in advance for any suggestions.

I am really not sure about what you are actually asking. But let me clear some things that i feel u should know.

1) Electron cannot travel faster than light due to relativistic effects. It can travel at 99% speed of light at tops. Not more than that.

2) You cannot use classical equations in this scenario to calculate acceleration cause they don't work with relativistic effects. If you use them, than you will get weird answers.

3) The relativistic effect that is involved in this scenario is the increase of mass of an electron as it approaches ‘c’. The closer it approaches to ‘c’, its mass will increase accordingly. Because of this increase in mass, an infinite amount of energy will be required to reach ‘c’ or exceed it.

4) Electron location does not effect its velocity or acceleration cause you are taking a homogeneous field.In a non-homogeneous field, the location will matter.

5) Electron speed can never be zero. Its just impossible for an Electron to have zero velocity.

5) Electron speed can never be zero. Its just impossible for an Electron to have zero velocity.
Whatever you meant to say, I don't think that's it. There is no difficulty in defining a frame in which an electron is at rest.

Whatever you meant to say, I don't think that's it. There is no difficulty in defining a frame in which an electron is at rest.

I meant practically speaking, an electron cannot be stationary i.e it cannot have v = 0 in a real system.

I meant practically speaking, an electron cannot be stationary i.e it cannot have v = 0 in a real system.
Why not? Clearly the speed of an electron can be zero in some frame.
Moreover, if you e.g. trap an electron in a Penning trap (which is constructed using the same "building blocks" as a SEM) the speed in the "lab frame" can be very ,very small (and the average speed will be zero).

I probably defined my "problem" at first with a lot more complexity than needed, since my last message I thought about a very short version which, after thinking, should have been the only one to start with:
By looking for elecron speed calculus, I thought I would have found equations that completely describe electron movements, I mean the speed and acceleration.
However, even though I usually find info and equations about speed, I haven't found any about acceleration.
So, beside the fact that I now can properly define the speed of an electron in a vacuum and under an uniform electric field, the way an electron gains its speed remains a mystery.

I have the feeling that something could be done by calculating the derivative of the speed equation but it is precisely the kind of thing I am not good at (yes, shame on me !).

Again, thank you for already given help and for any other coming one.

I probably defined my "problem" at first with a lot more complexity than needed, since my last message I thought about a very short version which, after thinking, should have been the only one to start with:
By looking for elecron speed calculus, I thought I would have found equations that completely describe electron movements, I mean the speed and acceleration.
However, even though I usually find info and equations about speed, I haven't found any about acceleration.
So, beside the fact that I now can properly define the speed of an electron in a vacuum and under an uniform electric field, the way an electron gains its speed remains a mystery.

I have the feeling that something could be done by calculating the derivative of the speed equation but it is precisely the kind of thing I am not good at (yes, shame on me !).

Again, thank you for already given help and for any other coming one.

The electron accelerates due to the force that is exerted on it when it is surrounded by a magnetic field.

However, even though I usually find info and equations about speed, I haven't found any about acceleration.
Surely though you've found one about force, namely the Lorentz force equation. Write ##F=\frac{dp}{dt}## where ##p## is the momentum ##\gamma{m}v##, and the chain rule will get you to an expression for ##\frac{dv}{dt}##, which is the acceleration.