Square Plate Parallel, with angle, Capacitance Problem

[sdhick]

Given two parallel plates, length a. One being on the x axis, starting at the origin. The other starting a distance, d up the y-axis at an angle theta (parallel to the x axis), for theta being small. I need to find an equation for an approximation of the capacitance.


I know that if the two plates were purely parallel, C = (E0 * A)/d = (E0 * a^2)/d and that the equation I'm looking for should be (E0 * a^2)/d *( something )
And I know that I need to think of this as a chain of small infinite capacitance in parallel along the angled plate. Meaning I need to do an integral.


I however do not even know where to start. I don't want anyone to just give me the something, but I would love some help on where to start with the integral.

I'm thinking it will have to be and integral (not sure on the limits) of the E field dotted with the ds(or distance).

I would appreciate any help you can give me. Thanks in advance!

 

[berkeman]

Given two parallel plates, length a. One being on the x axis, starting at the origin. The other starting a distance, d up the y-axis at an angle theta (parallel to the x axis), for theta being small. I need to find an equation for an approximation of the capacitance.


I know that if the two plates were purely parallel, C = (E0 * A)/d = (E0 * a^2)/d and that the equation I'm looking for should be (E0 * a^2)/d *( something )
And I know that I need to think of this as a chain of small infinite capacitance in parallel along the angled plate. Meaning I need to do an integral.


I however do not even know where to start. I don't want anyone to just give me the something, but I would love some help on where to start with the integral.

I'm thinking it will have to be and integral (not sure on the limits) of the E field dotted with the ds(or distance).

I would appreciate any help you can give me. Thanks in advance!

What a fun problem! I hadn't seen this one before.

You absolutely have the right idea -- that's how I would approach the problem. To help you set up the integral, start by drawing a small number of discrete caps, with the plates spaced farther and farther apart. I'd start with something like 4 cap plate pairs. Use that drawing to figure out what the y separation as a function of x is (it will be a trig function... which one?), and what the delta-x is (that relates to the area). Then see if that helps you set up the integral in the limit where delta-x shrinks to zero. Post your work here if it doesn't click for you and we'll help based on what you post.

Fun problem. Welcome to the PF.


EDIT -- an intermediate step if you need it, between the 4-cap drawing and the integral, would be to write a summation equation, showing the equation and the terms for the summation of those 4 capacitors.