Electron Movement (Integrating Over Nonuniform Electric Field)

[collegecolleg]

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?

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)

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!

 

[anathema]

Thank you for your interesting question. I would approach this problem by first identifying the relevant equations and variables involved. As you have correctly stated, Coulomb's Law and Newton's Second Law are the main equations to consider in this scenario. The variables involved are the charge of the electrons (q), their mass (m), the distance between them (r), and the time interval (t).

To simplify the problem, I would also assume that one of the electrons is fixed, as you have suggested, so that we can focus on the motion of the other electron. This means that the force acting on the moving electron is solely due to the fixed electron.

To calculate the force between the two electrons, we can use Coulomb's Law: F = kq1q2/r^2. Since both electrons have the same charge (q1 = q2), we can simplify this to F = kq^2/r^2. Given the charge of an electron (q = 1.6 x 10^-19 C) and the distance between them (r = 20 nm = 2 x 10^-8 m), we can calculate the force between them to be approximately 1.44 x 10^-27 N.

Next, we can use Newton's Second Law (F = ma) to calculate the acceleration of the electron. Since we are assuming no other external forces, the force we calculated above is equal to the force acting on the electron, and therefore, we can set F = ma. Rearranging this equation, we get a = F/m. Plugging in the values we have, we get an acceleration of approximately 9 x 10^17 m/s^2.

Now, to calculate the vertical distance the electron will move in the given time interval (t = 0.00001 s), we can use the kinematic equation delta x = 1/2at^2. Since we are only interested in the vertical distance, we can ignore the horizontal motion and focus on the vertical acceleration (a = 9 x 10^17 m/s^2). Plugging in the values, we get a vertical distance of approximately 4.5 x 10^-9 m.

Therefore, after 0.00001 seconds, the two electrons will be approximately 4.5 nm apart vertically. I hope this helps you with your problem. If you have any further questions or concerns