Airsteve0
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Homework Statement
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.
Homework 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])
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!