Application of Coulomb's law to positron velocities

[Aeonace32]

I have an idea in which I need to manipulate a positron using an electrical field. However, in order for the problem to work, I need to make sure that the positron's kinetic energy is under 511 keV. to do this, I used Coulomb's law (F=$\frac{kQ1Q2}{r^2}$) to obtain the force. Because F = ma, I divided both sides by mass to get acceleration. I know that integrating acceleration with respect to time yields velocity. However, in this case, acceleration is not a function of time, but rather a function of distance. My goal is to obtain an equation which can give me the velocity at different distances so that I can plug the right velocity into the kinetic energy formula. Is integrating the way to solve this, and if so, how do I do it correctly?

 

[Morgoth]

$\frac{dF(x(t))}{dt}=\frac{dF(x(t))}{dx} \frac{dx(t)}{dt}$

 

[Aeonace32]

Sorry, but could you elaborate a little?

 

[Morgoth]

$a=\frac{dv}{dt}=\frac{k}{m}\frac{{Q}_{1}{Q}_{2}}{r^{2}}$

what you write is this:

$\frac{dv}{dr} \frac{dr}{dt}=\frac{k}{m}\frac{{Q}_{1}{Q}_{2}}{r^{2}}$

dr/dt is the velocity so:

$v \frac{dv}{dr}=\frac{k}{m}\frac{{Q}_{1}{Q}_{2}}{r^{2}}$

 

[mfb]

You can directly get the energy by integrating the force over a distance (or using the Coulomb potential), you do not have to use any time-dependence at all for this.
As a bonus, you avoid relativistic effects in this calculation, which can influence the acceleration.

I need to make sure that the positron's kinetic energy is under 511 keV.

Are you sure this is the real requirement?

 

[Aeonace32]

Sorry if I'm starting to sound incompetent, but what would I do about the $\frac{dv}{dr}$

 

[Aeonace32]

@ mfb:
Thanks. The reason that I need to make sure it is under 511 keV is because if it is any higher than that, the positron-electron annihilation will start to produce heavy particles instead of energy.

 

[Morgoth]

$v \frac{dv}{dr}=\frac{k}{m}\frac{{Q}_{1}{Q}_{2}}{r^{2}}$

$\frac{1}{2} \frac{dv^{2}}{dr}=\frac{k}{m}\frac{{Q}_{1}{Q}_{2}}{r^{2}}$

$\frac{d}{dr} ( \frac{1}{2} v^{2}+\frac{k}{m}\frac{{Q}_{1}{Q}_{2}}{r} )=0$

$\frac{1}{2} v^{2}+\frac{k}{m}\frac{{Q}_{1}{Q}_{2}}{r}=const$

 

[Aeonace32]

Thanks for the information.

 

[mfb]

If you already have positrons, they can annihilate with electrons - and the cross-section is smaller for higher energy. Very high-energetic positrons might produce gamma rays of >1 MeV somewhere, which could do pair-creation, but as you can see this already requires more than 1 MeV of energy.

 

[Aeonace32]

I see. However, the main focus is not the amount of energy generated, but rather the efficiency of matter converting into energy.