Electrostatic Conformal Mapping Problem

[Airsteve0]

Homework Statement

The transformation z=1/2(w + 1/w) 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) = Φ(u,v)+iΨ(u,v) = (−λ/2πϵo)Ln(w) + Vo

x = 1/2(u + u/(u^2+v^2))

y = 1/2(v - v/(u^2+v^2))

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) (answer show above). 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!

 

[marcusl]

Cartesian coordinates are a poor choice since the natural symmetry of the problem is circular (think about what the electric field lines and equipotential surfaces look like in w). Change to circular polar coordinates instead. Write [itex]w=\rho\exp(i\phi)[/itex], and rearrange terms to get something that has a sum of terms involving [itex]\cos(\phi)[/itex] and [itex]\sin(\phi)[/itex]. In the z plane this gives you equipotentials that are confocal ellipses (circles in w map to ellipses in z). See if you can figure out what the field lines transform to.

Note that the line x<|1| is a branch cut.