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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.

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# I Electron speed and acceleration in an electric field

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