Electric force and free electrons

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In the discussion on electrostatic equilibrium involving two point charges of -1.0 micro C and +1.0 micro C, participants conclude that a free electron and a free proton cannot achieve equilibrium between the charges. The reasoning is that the forces acting on a test charge do not balance at any point along the meterstick. By applying the formula kq1q2/r^2, it is shown that the only potential equilibrium point is at x=0, where the forces from both charges act in the same direction and cannot cancel out. Therefore, no position exists for either a free electron or a free proton to be in electrostatic equilibrium. The discussion confirms the necessity of opposite charges for equilibrium to occur.
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Two point charges of -1.0 micro C and +1.0 micro C are fixed at opposite ends of a meterstick. Where could a) a free electron and b) a free proton be in electrostatic equilibrium?

I'm guessing the answer is nowhere since there have to be opposite charges to be in equilibrium?
 
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I think you are right. The forces don't balance anywhere.
 
is there any way I can prove it with the formula kq1q2/r^2?
 
Put one charge at r=(1/2) and another at r=(-1/2). Then the magnitude (ignoring direction) of the force on the third charge located at x due to one charge is k/(x-1/2)^2 and the other is k/(x+1/2)^2. If you set those equal you find the only possible position is x=0. But at x=0 you know the directions of both forces are the same. So they can't cancel.
 
yes! that's what I thought... thank you!
 

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