- #1

Spyro386

1. Homework Statement

1. Homework Statement

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.

## Homework Equations

$$\nabla^2 = 0$$

(U=constant at the boundary of the conductor)

$$U = -\frac{\lambda}{2 \pi \epsilon_0} ln(r)$$

## 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?