Electromagnetics: Laplace's Equation Cartesian

[karenmarie3]

Homework Statement

Two long metal plates of length L>>H (their height) intersect each other at right angles. Their cross section is a cross with each line of length H. This configuration is held at a potential V=Vo and the total charge up to a distance d (d<<H) from the center of the cross is given as q. Use separation of variables in 2D to find the near field potential for distances abs(x), abs(y) less than/equal to d from the center of the cross. How is the charge distributed from the center up to this distance d?

Homework Equations

In class we solved this problem for one 'quadrant' of the open space of the cross:
V = -[Q/(εoL^2)]xy+Vo

By that I mean the rest of the cross was not there, just 2 conducting planes connected at 90 degrees.

The Attempt at a Solution

I was thinking I could use principle of super position here, combined with the fact that if I set the line where the 2 metal plates as my 'z' axis, that my signs of x and y will alternate as I go around the cross. This gave me a result of 4Vo when I added them all together. I'm not entirely sure that is right. Also my professor mentioned something about the sign of the electric field changing from 'quadrant' to 'quadrant'... this also would seem to suggest I have made an error in my thinking.

Any suggestions or tips?

 

[mark726]

Thank you for your post. I would like to offer some suggestions for solving this problem.

Firstly, I would recommend using the method of separation of variables to solve this problem. This involves breaking down the problem into simpler parts and solving them separately. In this case, you can consider the potential at different points along the x and y axes separately.

Secondly, I would suggest considering the electric field instead of the potential. The electric field is related to the potential by the equation E = -∇V, where ∇ is the gradient operator. This will make it easier to visualize and understand the distribution of charge.

Thirdly, I would recommend carefully considering the boundary conditions for this problem. The potential at the intersection of the two plates must be continuous. This means that the potential on one plate must be equal to the potential on the other plate at the point of intersection.

Lastly, I would suggest checking your calculations and considering the physical implications of your solution. For example, does your solution make sense in terms of the distribution of charge on the plates and the electric field lines?

I hope these suggestions will help you in solving this problem. Good luck!