What Trajectory Would a Proton Follow Near a Fixed Positively Charged Particle?

[Nabeshin]

I know it's relatively easy to show that the only possible orbits given an inverse square centripetal force relation are conic sections, of which the only stable orbits are ellipses. However, I was wondering what the situation would be given a centrifugal force (really a misnomer, since it wouldn't follow anything like a circular trajectory) but basically something in a situation like this:

Fixed positively charged particle and proton flying nearby.

In this situation the force is always anti parallel to the radius vector. Obviously there is no bound orbit for this situation, and I'm thinking the type of path is going to be highly sensitive to initial conditions, but before I waste time working over the problem I figure I'd ask if any of you have insight into it (or have done anything similar).

Cheers!

 

[csprof2000]

Due to the inverse square relationship, I believe it will still be a conic section - in this case, the only sensible one - a hyperbola.

In fact, a hyperbola makes good physical sense here. Far away, the thing moves in a straight line, then when it collides it has a large curvature, then it quickly gets back down to "asymptotic", straight-line behavior.

That's my guess.

 

[astrokreng]

I would like to clarify that the concept of centrifugal force is a fictitious force and does not actually exist as a physical force. It is simply an apparent force that arises due to the inertia of an object in motion and the frame of reference in which it is observed. In the scenario described, the force acting on the proton would be the electromagnetic force between the positively charged particle and the proton, which is a real force.

In terms of the trajectory of the proton, it would indeed be highly sensitive to initial conditions due to the absence of any stable orbit. The path of the proton would depend on the relative strength of the electromagnetic force and the initial velocity and position of the proton. It is possible that the proton could follow a spiral path or even escape the influence of the fixed particle altogether.

I would also like to mention that the concept of bound orbits applies to gravitational forces, which follow an inverse square relation. In the case of electromagnetic forces, the force relation is different and thus the types of orbits that are possible may also be different.

In conclusion, while it is an interesting thought experiment, the scenario described does not fall under the dynamics of centrifugal force and would require a different approach to analyze the trajectory of the proton.