# Electron Movement (Integrating Over Nonuniform Electric Field)

#### collegecolleg

1. Homework Statement
Two electrons with 20eV are 20 nm a part are move horizontally. Ignore relativistic effects and assume no other external forces.

How far apart are they vertically after .00001 seconds?

2. Homework Equations
Coulombs Law (the force between the two electrons) F=Kq1q1/r^2
F=ma (to calculate the acceleration where m is the mass of the electron)
delta x = 1/2at^2 (no initial vertical velocity, can calculate the change in distance)

3. The Attempt at a Solution

So I can calculate the force between them using coulombs law, the acceleration by using F=ma, and then given a time, the change in x using the kinematic equation. I am pretending that one of the electrons is fixed so that I can just calculate how far one moves (which I am not sure if that is correct way to do it or not).

The problem is that as the electron moves away the force decreases (proportionally to r^2). So I can divide the time up into a bunch of time slices and recalculate the distance it moves with olddistance += 1/2 * kq1q2/(m*olddistance^2)*timestep^2, but this doesn't really get me a close form, and I am afraid that floating point precision is going to mess with my results.

Thus is there some way for me to integrate it and get a closed form. I think maybe I can setup the electric field and kind of do a line integral or something...

Thanks for any help!

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