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Electrostatic Conformal Mapping Problem

  1. Mar 21, 2012 #1
    1. The problem statement, all variables and given/known data
    The transformation [itex]z[/itex]=[itex]\frac{1}{2}[/itex]([itex]w[/itex] + [itex]\frac{1}{w}[/itex]) maps the unit circle in the w-plane into the line −1≤x≤1 in the z-plane.

    (a) Construct a complex potential in the w-plane which corresponds to a charged
    metallic cylinder of unit radius having a potential Vo on its surface.

    (b) Use the mapping to determine the complex potential in the z-plane. Show that
    the physical potential takes the value Vo on the line −1≤x≤1. This line thus
    represents a metallic strip in the x-y plane.

    2. Relevant equations

    F(w) = [itex]\Phi(u,v)[/itex]+i[itex]\Psi(u,v)[/itex] = [itex]\frac{-\lambda}{2\pi\epsilon_o}[/itex]Ln(w) + Vo

    x = [itex]\frac{1}{2}[/itex](u + [itex]\frac{u}{u^2 + v^2}[/itex])

    y = [itex]\frac{1}{2}[/itex](v - [itex]\frac{v}{u^2 + v^2}[/itex])

    3. The attempt at a solution

    So far I have worked out the relation between (x,y) and (u,v) as well as made an attempt at part (a). However, it is part (b) and using the mapping that I am completely lost with. Mainly, if I try to find u and v in terms of solely x and y I get 2 solutions (i.e. plus or minus because of squaring); this leaves me unsure of what to do. Any help would be wonderful!
     
  2. jcsd
  3. Mar 22, 2012 #2

    marcusl

    User Avatar
    Science Advisor
    Gold Member

    Please do not double post. I have replied to the version in Homework/Advanced Physics.
     
    Last edited: Mar 22, 2012
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