# Voltage,electron energy and electrode spacing

I am trying to calculate what is theoreticaly maximum energy electron will gain between two electrodes.The electrodes are 100 micron apart and one is +50V,and other is -50V.

I learned that eV unit equals electron being accelerated across potential of 1V,but electrons cant accelerate instantly since they have mass.In the ideal scenario that the electrons begins at the surface of negatively charged electrode a fly straight towards positive electrode,will it gain 100 eV considering the gap is only 100 micron?

How fast can the electron accelerate,isnt the 100 micron gap with 100V potential just too small to give the electron the full 100eV energy? I mean consider the extreme case,that electron is between electrodes 1 nm apart and there is potential 1 giga volts,the electron wouldnt go from being stationary to 1 giga eV energy in 1 nano meter distance.

How can I calculate if the electron can gain the full potential energy in specific gap distance and voltage combinations? I want to look at gap distance,voltage,and tell how much energy can electron have in those gap spaces.

I would like to have some kind of rule of thumb in my head,something like a electron will go from zero to 1000 km/h in 1mm gap with 100V potential in 1 microsecond or something like that.

• Electrons can be considered massless for most practical purposes. Theoretically just using the Newtonian mechanics should get you in the ballpark. – safesphere Sep 21 '17 at 19:58
• But, with respect, in order to apply the Newtonian mechanics to an electron, you need to treat it as having a mass! – Philip Wood Sep 22 '17 at 11:55

## 1 Answer

It will always be the case that as the electron moves across the gap:$$\text{KE gained = PE lost} \ .$$So if the electron starts from rest$$\frac{1}{2}m v^2=e \Delta V$$If the electron travels from a negative to a positive electrode with a given pd between them, the separation of the electrodes doesn't affect the final speed of the electron. What you suggest in your third paragraph simply isn't the case.

A rule of thumb that I use is that a pd of 2500 V will accelerate an electron initially at rest up to approximately $\frac{1}{10}$ of the speed of light.

At this energy (2500 eV) there is an error of about $\frac{1}{2}$ % in using the $\frac{1}{2}mv^2$ formula for kinetic energy. The error gets worse and worse at higher energies and you need to use the relativistic formula $$KE=(\gamma - 1)m c^2$$ in which $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.$

The time that the electron takes to go from one electrode to the other does depend on the separation and shape of the electrodes. The easiest case is when the electrodes are plane and parallel and the gap between them is small compared with their length and breadth (or diameter). In that case the field between them is uniform and the acceleration of the electron is constant, so the time taken is plate separation divided by the mean velocity of the electron, which is half its final speed if it starts from rest on the negative electrode.

• I dont understand equations,can you explain it more simply? Car with certain weight ( electron mass ) and horsepower ( voltage ) accelleration and top speed ( eV ) in dragstrip of certain lenght would be good way to explain it to me. – wav scientist Sep 21 '17 at 21:14
• Concentrate on what I said in my comment up to the rule of thumb. It's pretty equation-free up to that point. – Philip Wood Sep 21 '17 at 21:25
• But how fast will it accelerate? How many nanoseconds? And over what distance will it accelerate from start to final velocity? – wav scientist Sep 21 '17 at 21:27
• Its final speed is controlled by the potential difference through which it has moved, and is nothing to do with the distance gone. In a uniform field the time taken to get to a certain speed is found by dividing the distance it has travelled in the field (after it started from rest) by its mean speed, that is half its final speed. – Philip Wood Sep 21 '17 at 21:39
• So back to my example,two electrodes,one +50V,one -50V,100 micron gap.Electron starts at rest at the surface of negative electrode. 1. How long does it take to reach final speed? 2. What distance will it travell before it reach final speed? – wav scientist Sep 21 '17 at 21:49