Method of Images Question (Concave Geometries)

In summary, the problem with image charges on curved conducting plates is that they cannot be solved with images. A hyperbola might be a reasonable approximation for the catenary shape, but this cannot be confirmed.
  • #1
^_^physicist
235
1
I've been working with image charges for a while now, and I have noticed that I haven't been able to find a single discussion on the use of image charges for concave geometries (i.e. a charge on the "inside" of a catenary shaped curved conducting plate, see attached picture). Has anyone worked with a problems of this nature before? Anyone know of any resources that discuss problems of this nature? I have been racking my brain for a over a few days and I can't seem to figure out how to a approach a problem like this. I almost just broken down and tried to solve these geometries with boundary conditions and the laplace/possion equations (depending on the problem), but they become a mess way to quickly.

Any help is appreciated.

(A note on the picture, the point on the right is a negative charge and the point on the right is a positive charge. I am very concerned with this problem for a research project; however, I would be just as happy (in fact more so) if someone could give an explanation for solving the simplified problem of just looking at the left-hand side (or right-hand side) of the graph)
 

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  • #2
Most geometries cannot be solved with images. Cylinders, planes, wedges and spheres are the cases that are known. If your curve possesses a transformation to a line or circle (I doubt catenaries qualify), you might have a chance using conformal mapping techniques. Weber's book Electromagnetic Fields covers this, also Smythe.
 
  • #3
Well a semi-reasonable approximation to the catenary shape is that of a hyperbola (at least at distances close enough to the hyperbola, and since my problem is with something close enough to the catenary anyway, I am not too concerned with making this substitution).

I don't have much experience (i.e. no experience) with conformal mapping, so would a hyperbola be a something we could work with? Heck would any conic section be something that would work well with conformal mapping. (Also, thanks for the recommendation on the Weber book, I am picking it up today).
 

Related to Method of Images Question (Concave Geometries)

1. What is the Method of Images?

The Method of Images is a mathematical technique used to solve boundary value problems in electrostatics and fluid dynamics. It involves creating a virtual image of a charge or object in a system to account for the effects of boundaries or other objects in the system.

2. How is the Method of Images used in concave geometries?

In concave geometries, the Method of Images is used to determine the electric potential and field at points inside or outside of a concave conductor. This is done by creating a virtual image charge outside the conductor to balance the real charge inside.

3. What are the advantages of using the Method of Images in concave geometries?

The Method of Images allows for an analytical solution to be obtained for the electric potential and field in complex concave geometries. This can save time and resources compared to other numerical methods.

4. Are there any limitations to using the Method of Images in concave geometries?

Yes, the Method of Images assumes that the conductor is infinitely thin and that the charges are in a uniform medium. Additionally, the method may not be applicable to highly irregular geometries.

5. Can the Method of Images be used for other geometries besides concave?

Yes, the Method of Images can also be applied to convex geometries, as well as other types of boundary value problems in electromagnetics and fluid dynamics. However, the specific approach may differ depending on the geometry and problem at hand.

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