Force Between Moving Charge Particles: Does Speed Matter?

[Chris Frisella]

You have two electrons held at a fixed distance X from each other. You measure the opposing force is Y. Now you bring the electrons closer to each other, and then let them go so they can fly apart. As the electrons move apart and reach distance X, will the opposing force still be Y, or will it be less/more than that? Similarly, if you reserves it so you had the particles approaching each other (let's say electron & positron pair), what would happen with the force between them at distance X?

Put simply, does the speed towards or away from each other effect force?

 

[Dale]

Yes, the speed does affect the force. If you want to calculate this the Lienard Wiechert potentials will let you calculate the fields, and then the Lorentz force will give you the force.

So, in units where c=1 and q=1, if you have a charge A at x=-1 and a charge O at the origin then the force on O is f=(1,0,0). Now, if charge A is moving uniformly towards the origin at v=0.6 then f=(0.64,0,0). In that moving case, because you specified motion exactly towards the other charge there is no magnetic field at O, so the force on O does not depend on the speed of O, only the speed of A. (That is not a general result, but a very specific result only for the geometry specified)

 

[Chris Frisella]

Thanks! I think I get the math, but I'm a little unfamiliar, so if the particles were moving apart at the same rate in your example, the reduction in force would be the same as well?

 

[Dale]

Yes, it works out the same, but again that is dependent on the geometry specified. If they are not moving on a direct line then towards or away can make a difference.

 

[Chris Frisella]

Ok, cool! Now comparing them moving apart vs moving towards each other, but in both cases not on a perfectly direct line (just slightly off). If my understanding of the fields is correct, the moving apart scenario wouldn't experience as much a decrease in force as the moving towards, yes?

 

[Dale]

At this point you may want to try working this out for yourself. Do you have any math software? If so, then you can write a little routine to calculate the Lienard Wiechert potentials. If not, then use pencil and paper but look for very simple scenarios.

 

[Chris Frisella]

Was just doing that :) SO actually it looks like I had that backwards, yea?

 

[Dale]

That is good! I will have to check later, I don't know that result off the top of my head so I will have to run it through my own routine.

 

[Chris Frisella]

Cool, thanks! I'm interested to see what you get!

 

[Dale]

OK, so for O at the origin A at (-1,0,0) and for velocity of O = v and velocity of A = -v with v = (.6,.1,0) I get f=(0.633174, 0.0383742, 0.) and with v=(-.6,.1,0) I get f=(0.633174, -0.0383742, 0.)

 

[Chris Frisella]

So, no difference. Do you know if that aligns with empirical data?