# Electron trajectory

Ok, so this is a problem I made up, but it has been bothering me, so here it is!

## Homework Statement

There are two flat rectangular conductors of negligible thickness, both with a length of L and width of W. Each is charged up to a certain different voltage, V1 and V2 respectively. The conductors are lying parallel to each other in the same plane (i.e. imagine 2 flat pieces of metal lying on a table, and then place two edges adjacent and parallel to each other). These two edge-parallel conductors are separated by a distance P. The two plates' bottom edges are parallel to the x-axis and lay on the x-axis. Given that a particle with charge Q and mass M is placed a distance D from the left conductor’s edge and a distance H from the bottom edge of the conductor, give the equation for the path that the particle will take as it arcs from one plate to another. Assume that this occurs in a vacuum and that the left conductor’s voltage is equal in sign to that of the particle.

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OK, since I have no clue how to do this, I'll just ask some questions that I came up while looking at this problem.

1. Everything would be awesome if I could calculate the magnitude and direction of the force that the plate exerts on a charged particle given the coordintes, mass, etc. of the particle. But then, I remember that I've never learned how to calculate this! We've done vaguely similar problems like 'find the speed a particle is at an infinite distance away, blah, blah' but never when the particle was a certain distance away! So then, I take a look in the book and they mention using charge density to find the force or voltage a certain distance away and then using calculus to sum it up. But then my question is

Does a general voltage correspond to a similar charge density?

I was thinking it should, since voltage occurs when electrons or a lack of builds up on a certain area relative to a ground reference. Eventually, I thought that since voltage is also defined as potential energy per charge, and that a negative voltage is due to a buildup of electrons (right?), that all one had to do to find the charge density was divide the voltage by the energy required to move an electron from the plate to infinity (infinity assumed 0 volts?). This would determine the amount of electrons present, and from there the charge could be determined. But thinking back, this would infer that no matter what the size of the plate is, the amount of electrons would always be the same, since nowhere did I mention taking conductor dimensions into account.

I'm a little confused here, though. I would think voltage and charge would be related, but I don't know... So anyone have help?

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## Answers and Replies

Hmm... I looked in the advanced physics page and came across a post called 'weird plate capacitor.' In it, it mentioned the 'image method', and I googled it and came across wikipedia, which stated

"The simplest example of a use of this method is that in 2-dimensional space of a point charge, with charge +q, located at (0, a) above an 'infinite' grounded (ie: V = 0) conducting plate, lying along the x-axis. Deriving any results from this setup, such as the charge distribution on the plate, or the force felt by the point charge, is not trivial."

So, it seems I'm going to have to learn this concept first. I will figure it out eventually, I assure thee!