# Accelerating an electron in beta decay

• A
• Cato
In summary, the electron created and emitted in the beta decay of a proton has an initial velocity close to the speed of light. When I try to calculate, not taking into account relativity, the force needed to accelerate an electron to that velocity over a distance the size of a proton, I get about 45 N. That seems absurdly high and I can't believe that it is right.

#### Cato

The electron created and emitted in the beta decay of a proton has an initial velocity close to the speed of light. When I try to calculate, not taking into account relativity, the force needed to accelerate an electron to that velocity over a distance the size of a proton, I get about 45 N. That seems absurdly high and I can't believe that it is right. What errors in math or in my fundamental understanding of physics am I making?

The electron does not need a force to get accelerated, it starts with a high speed.

Unrelated (!): if you calculate the force between two elementary charges separated by 1 femtometer, you get ~200 N.

Well then, thanks. So is was that my math was ballpark OK but my assumptions about physics were off.

I'll do you the courtesy of responding to you like a physicist (brace yourself):
Your physics was way off. Your maths was based on the physics - therefore the maths was irrelevant.
Your inuition that 25N "seems absurdly high" for nuclear reactions is false... it was about an order of magnitude too small.
Don't sweat it: there is nothing wrong with getting things wrong - you did what you could with the understanding you had, and had the sense to question the results ... that's better than most people manage (most don't even make the attempt) so well done.

Recap:
Beta particles are not created at rest and then accelerated: they are created with kinetic energy.

note: You work out the kinetic energy from the mass differences using ##E=mc^2##

Ah, yes, thanks. I have braced myself. I do appreciate your response. Yep, the magnitude of the strength of the electromagnetic force, or of any kind of description of fundamental physics or discussion of extremely large numbers, is far beyond the sort of "common sense" approach I reacted with. The universe is a wonderful place.

Simon Bridge said:
Beta particles are not created at rest and then accelerated: they are created with kinetic energy.

I understand this, but wouldn't plus charged particles get accelerated, too? I mean, how can they not, if they are so close to the nucleus?
I don't understand when you say 250Ns of force would cause the acceleration, but it somehow does not?
So, are you implying that the emitted positron (for example) is never that close to the nucleus? (Like 1 fm close)
Or the force actually acts on the particle, but for a such short period, that it doesn't have a extreme effect?

The de-Broglie wavelength for a 1-2 MeV electron is of the order of hundreds of fm. Trying to localize the electron in the decay better than that does not make sense.

Garlic said:
I mean, how can they not, if they are so close to the nucleus?
The beta+ and beta- of course get interactions with the nuclei coulomb's potential... This is for example, the reason why the electron and positron momentum spectra are not identical.
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/imgnuc/betapcu64.gif
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/beta2.html#c1
Then again, the transition is like a 3 body decay, so the momenta of the beta particles can vary a lot [given the phase space they have free from the transition's released energy]...As a result there can be beta particles with higher energy and beta particles with low energy... the electrons however do get swifted to lower energies than the corresponding positrons would because some amount of energy is needed to overcome the attractive potential [and allow the transition].

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