Electric Charge Image Theory - Half-Infinite Planes


by aphid
Tags: charge, electric, halfinfinite, image, planes, theory
aphid
aphid is offline
#1
Mar11-10, 08:14 PM
P: 2
1. The problem statement, all variables and given/known data
A point charge Q lies along the axis of symmetry between two half-infinite grounded conducting plates which form an angle theta (less than or equal to 90 degrees).
a. What are the magnitudes and locations of the image charges?
b. What is the total charge on the conducting plates?
c. Where does the charge come from?


2. Relevant equations
An application of Laplace's Equation I believe . . .


3. The attempt at a solution
I'm at somewhat of a loss here. The image of a point charge situated above a single infinite plane is simply its negative mirror image, resulting in an equivalent voltage field above the plane. My intuition tells me that I can somehow use superposition of the infinite plane case, but it would be a shot in the dark. I'm also fairly certain that a total charge -Q accumulates on the conductor, but I don't know how to explain this.

Any insight into this problem would be greatly appreciated.
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marcusl
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#2
Mar11-10, 09:37 PM
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P: 2,020
Start with finding the image locations by geometric construction. The first image is just the usual one you are familiar with (times two). But there's an image of each image in its opposite sheet. And so on. You'll have an infinite series.
aphid
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#3
Mar11-10, 10:25 PM
P: 2
Not sure if I am doing this correctly (see attached image). Am I supposed to extend each plane, effectively creating two intersecting infinite planes? It gets pretty messy once I start projecting across the 'imaginary' portions of the planes.

Attached Thumbnails
000689.jpg  

marcusl
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#4
Mar12-10, 10:21 AM
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P: 2,020

Electric Charge Image Theory - Half-Infinite Planes


Yes, you've done this correctly (more or less--some of your distances don't match), and wow it gets messy. I tried it myself and found for the angles I used that the number of images was finite, so I looked it up. If the angle between planes is pi/n where n is an integer, there are (2n-1) images. In general, however, the number is infinite as I wrote above.

I agree that the total induced charge should be -Q.


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