Motion under repulsive electrostatic force

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Discussion Overview

The discussion revolves around the motion of two particles under a repulsive electrostatic force, exploring the nature of their trajectories and the mathematical framework governing their motion. Participants consider the similarities and differences between electrostatic and gravitational forces, particularly in terms of the equations of motion and the resulting paths of the particles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire whether there is a simple curve that two particles follow under repulsive electrostatic forces, similar to gravitational forces.
  • It is noted that the force between like charges is described by Coulomb's law, which states that the force is proportional to the inverse square of the distance between the charges.
  • One participant mentions that in the center of mass (cm) system, each particle follows a hyperbola, with the differential equations being analogous to those for attractive forces but differing in sign due to the nature of the charges.
  • There is a suggestion that if one particle is fixed, it occupies a focus of the hyperbola traced by the other particle, differing from the case of attractive forces.
  • Concerns are raised about the time parameterization of the motion, with one participant expressing uncertainty about how to express time as a function of distance after separating variables.
  • Another participant suggests using center of mass variables if neither particle is fixed and recommends consulting advanced mechanics texts for further details.

Areas of Agreement / Disagreement

Participants express a range of views on the nature of the motion under repulsive forces, with some agreeing on the hyperbolic trajectory in the center of mass system, while others remain uncertain about specific aspects such as time parameterization. No consensus is reached regarding the overall understanding of the motion.

Contextual Notes

The discussion highlights limitations in the participants' understanding of the time parameterization and the mathematical derivations involved, with references to advanced mechanics literature for further exploration.

lark
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Is there a simple curve that 2 particles follow when there's a repulsive electrostatic force - like there is for gravitational forces?

I don't know how to solve the differential equation that you get for the
motion.

Laura
 
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When two particles have the same polarity (either both are + charged or - charged), they repel (rather than attract if charges are opposite) each other. The force is proportional to the inverse of the square of the distance (similar to the gravitational force law), and is described by Coulomb's law.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html

Using F = ma = m\ddot{x}, where x is relative to some reference point, e.g. the distance from one charge to the midpoint between the charges, which is then half of the distance between the charges. So there must be an equation of motion for each charge, for which the force is common, and which varies according to the separation of the charges.

If the charges have different masses, then they will have different accelerations, but the force one is the same as the other, even if the charges are of different magnitude.
 
lark said:
Is there a simple curve that 2 particles follow when there's a repulsive electrostatic force - like there is for gravitational forces?

I don't know how to solve the differential equation that you get for the
motion.

Laura
In the cm system, each particle follows a hyperbola.
The DE's are the same as for attractive force (like gravity), but with a different sign for qq'. The full derivation is in advanced mechanics books (like Goldstein), in connection with the classical derivation of the "Rutherford Scattering" cross section.
 
Meir Achuz said:
In the cm system, each particle follows a hyperbola.
The DE's are the same as for attractive force (like gravity), but with a different sign for qq'. The full derivation is in advanced mechanics books (like Goldstein), in connection with the classical derivation of the "Rutherford Scattering" cross section.
Yes, if one particle is fixed in place, it's at a focus of the other
particle's hyperbola! At the other focus than the one it would be at if the force were attractive.

But I don't know what the time parameterization looks like, though I separated the variables to get an integral for time as a function of distance.

Laura
 
lark said:
Yes, if one particle is fixed in place, it's at a focus of the other
particle's hyperbola! At the other focus than the one it would be at if the force were attractive.

But I don't know what the time parameterization looks like, though I separated the variables to get an integral for time as a function of distance.

Laura
If neither particle is fixed, you can use center of mass variables.
I suggest you look into Goldstein for more detail.
 

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