Electron Trajectory: Questions & Answers

[Durato]

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. ----
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?

 

[Durato]

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!

 

[Moetasim]

I would approach this problem by first understanding the basic principles of electricity and magnetism. The force exerted on a charged particle in an electric field is given by the equation F = qE, where q is the charge of the particle and E is the electric field strength. The direction of the force is determined by the direction of the electric field.

In this scenario, we have two charged conductors creating an electric field between them. The particles will experience a force in this field and their trajectory will depend on the direction and magnitude of this force. To calculate the electric field at a specific point, we can use the equation E = V/d, where V is the voltage and d is the distance from the conductor.

In terms of your question about charge density, voltage and charge are indeed related. Voltage is a measure of potential difference between two points, and this potential difference is created by a difference in charge between the two points. However, the charge density of the conductors will also play a role in determining the electric field and the resulting force on the particles.

To solve this problem, we would need to use the equations for electric field and force to determine the trajectory of the particles. We would also need to take into account the dimensions of the conductors and the distance between them. This can be a complex calculation, but it is possible to solve using mathematical techniques such as integration.

Overall, understanding the basic principles of electricity and magnetism is crucial in solving this problem. It would also be helpful to have a good understanding of vector calculus and mathematical techniques for solving complex equations.