# Electrostatic Conformal Mapping Problem

1. Mar 21, 2012

### Airsteve0

1. The problem statement, all variables and given/known data
The transformation $z$=$\frac{1}{2}$($w$ + $\frac{1}{w}$) 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.

2. Relevant equations

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

x = $\frac{1}{2}$(u + $\frac{u}{u^2 + v^2}$)

y = $\frac{1}{2}$(v - $\frac{v}{u^2 + v^2}$)

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. Mar 22, 2012

### marcusl

Please do not double post. I have replied to the version in Homework/Advanced Physics.

Last edited: Mar 22, 2012