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Potential near a conducting strip (Conformal maps)

  1. Aug 11, 2017 #1

    1. The problem statement, all variables and given/known data

    We have an infinitely long wire with charge density ##\lambda## located at ##x=0,y=h##. We also have a semi-infinite strip of a conductor ##|x|<a/2, y<0##. We need to find the potential in all space. The hint is to use conformal maps.

    2. Relevant equations

    $$\nabla^2 = 0$$
    (U=constant at the boundary of the conductor)
    $$U = -\frac{\lambda}{2 \pi \epsilon_0} ln(r)$$

    3. The attempt at a solution

    The general idea is to conformal map the strip to a more simple geometry where I could use the method of image charges. I've had two attempts at doing that:

    In first attempt I've simply first used the map ##f(z) = z*i## to rotate the strip and then using ##g(z) = \frac{z+a/2i}{z-a/2i}(-ia/2)## (sending -ia/2 to 0, 0 to ia/2 and ia/2 to inf) effectively strecthing the strip to the first quadrant. Finally I wanted to use ##h(z) = z^2## to map the first quadrant to upper half plane. I must've visualised ##g(z)## the wrong way as this map mapped my wire ##z=ih## inside the upper half plane.(When it was originally outside the conductor)

    The second attempt started in a similar way. I use ##f(z) = zi## to rotate the strip. Then I use ##g(z) = e^{\pi z/a}## to map that to ##{|z| >1 , Re(z)>0}## Using ##h(z) = z^{-2}## I map this to the unit disc and then using ##i(z) = \frac{z+i}{z-i}(-1)## I map the disc to the upper half plane again. The problem this time was the use of the exponential fuction. It made my potential peroidic(which looking at the original problem it shouldn't be), since exponential is periodic in complex.


    Where did my thinking go wrong in these steps? Could I avoid the use of exponential functions and still map to a simple enough geometry to solve analitically somehow?
     
  2. jcsd
  3. Aug 16, 2017 at 7:00 PM #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. Aug 16, 2017 at 9:22 PM #3

    haruspex

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    Is it periodic within the original strip, or only if you project the solution beyond the original strip?
     
  5. Aug 19, 2017 at 12:01 PM #4
    The strip is a piece of a conductor and there is a wire next to it. Isn't potential constant within the conductor? And I need to find the potential outside the strip.
     
  6. Aug 19, 2017 at 3:42 PM #5

    haruspex

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    You only provided the bare outline of your method beyond all the mappings, so it is not clear to me how you got your potential function. You mentioned method of images, so maybe you meant only the contribution to the potential from the charge distribution within the conductor.
    If you meant the actual potential in the conductor, I don't see how you could have got that. That it is constant would be the boundary condition you were plugging in.
    Maybe if you post more details it will become clear.
     
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