Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Method of Images Question (Concave Geometries)

  1. Feb 16, 2009 #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)
     

    Attached Files:

  2. jcsd
  3. Feb 16, 2009 #2

    marcusl

    User Avatar
    Science Advisor
    Gold Member

    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.
     
  4. Feb 16, 2009 #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).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Method of Images Question (Concave Geometries)
  1. Method of Images (Replies: 3)

Loading...