Electric Charge Image Theory - Half-Infinite Planes

In summary: This is because the system is symmetric about the xz-plane and the charges will cancel out along that plane. In summary, the problem involves a point charge Q between two half-infinite grounded conducting plates at an angle theta. The image charges can be found using geometric construction, resulting in an infinite series of images. The total charge on the plates is -Q, due to symmetry about the xz-plane. The problem can be approached using superposition of the infinite plane case.
  • #1
aphid
2
0

Homework Statement


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?


Homework Equations


An application of Laplace's Equation I believe . . .


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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  • #2
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.
 
  • #3
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.

http://photoanon.com/viewer.php?file=zasljrfnro9t0u4u1tm6.jpg
 

Attachments

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  • #4
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.
 

1. What is the Electric Charge Image Theory?

The Electric Charge Image Theory is a concept in electromagnetism that explains how electric charges interact with each other. It states that when a charged object is placed near a conducting surface, an image charge of the same magnitude but opposite sign is induced on the surface. This image charge creates an electric field that interacts with the original charge, resulting in a net force.

2. What are Half-Infinite Planes in relation to Electric Charge Image Theory?

Half-Infinite Planes are conducting surfaces that extend infinitely in one direction and have a boundary at the other end. They are often used in the context of Electric Charge Image Theory as they provide a simple and idealized case for studying the effects of the image charge.

3. How is Electric Charge Image Theory applied in real-life situations?

Electric Charge Image Theory is commonly used in the design and analysis of electronic devices, such as capacitors and sensors. It also has applications in electrostatics, such as the behavior of lightning rods and the attraction of dust particles to charged surfaces.

4. What are the limitations of Electric Charge Image Theory?

Electric Charge Image Theory is a simplified model and does not take into account the complex behavior of charges in real-life situations. It assumes that the conducting surface is perfectly smooth and that the charges are point-like, which may not always be the case. Additionally, it only applies to static charges and does not consider the effects of moving charges.

5. Are there any other theories that explain the interaction of charges with conducting surfaces?

Yes, there are other theories, such as the Method of Images and the Boundary Element Method, that can also be used to analyze the behavior of charges near conducting surfaces. These theories take into account more complex factors, such as the shape and size of the conducting surface, and can provide more accurate results in certain situations.

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