Equation for the Motion of a Charged Particle in an Electric Field

[BOYLANATOR]

Homework Statement

This is a problem I was thinking about which I haven't come across in my course.

Two particles with mass, m, and a positive charge, q, are separated by a distance r₀. One particle is unable to move while the other is released and is repelled. Find an equation for the motion of the particle.

Homework Equations

F=q²/cr²

a=F/m

The Attempt at a Solution

The problem I have in working out the acceleration at any time here is that r is a time dependent variable.
Although this seems like a pretty important thing to know how to calculate I end up going around in circles with the maths.

 

[voko]

This is not a simple problem. Even if we assume that the accelerating particle does not radiate energy (which in reality it would do), we still end up with a differential equation. You can get this equation from conservation of total energy. The resultant equation can be integrated, but it will be a complicated expression, where the distance depends on time implicitly.

 

[gneill]

You can easily find the velocity at some position r via the change in potential energy that occurs in moving from position r_o to new position r. That'll give you some expression:

$v = \sqrt{f(r)}$

where f(r) is a function of r (it's for you so work out!).

Then make v = dr/dt, shuffle to group the appropriate variables and integrate both sides. That'll leave you with an expression of the form t = g(r), for g some function of r (again, for you to work out).

As for finding r in terms of t, that is, r(t), well you might find that solving t = g(r) for r is not trivial. Good luck!