How to Calculate Likelihood of Electron Jumping a Gap in Air?

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To calculate the likelihood of an electron jumping a gap of 100 to 200 nm in air, one must consider the emitted electron count and the potential difference of 10 to 20 V. The formula N = N_0 e^(-?) is suggested, but the exponent needs to be determined for accurate calculations. The mean free path of electrons in standard temperature and pressure (STP) air is crucial, as it varies with electron energy. Understanding these factors will help in estimating the probability of electron gap crossing. The discussion emphasizes the importance of both the mean free path and the energy of the electrons in this calculation.
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I'd like to calculate the likeliness of an electron jumping a gap of 100 to 200 nm. I have a source emitting electrons, the potential is 10 - 20 V. What is the proper way of calculate this?

\displaystyle N = N_0 e ^{-?} where N_0 is the amount of emitted electrons. If I use this way, how do I determine the speed of the decline, in other words the exponent in the forumla?
 
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You may want to look up the mean free path of an electron in a STP air.

Note that the mean free path may also be dependent on the energy of the electrons.

Zz.
 

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