# Electrostatic Conformal Mapping Problem

• Airsteve0
In summary, 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. To determine the complex potential in the z-plane, it is recommended to use circular polar coordinates instead of Cartesian coordinates due to the circular symmetry of the problem. By writing w=\rho\exp(i\phi), the potential can be rearranged to involve terms in \cos(\phi) and \sin(\phi), resulting in confocal ellipses in the z-plane. However, the line x<|1| is a branch cut and should be taken into consideration when determining the field lines in the z-plane.
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!

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 $w=\rho\exp(i\phi)$, and rearrange terms to get something that has a sum of terms involving $\cos(\phi)$ and $\sin(\phi)$. 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.

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## 1. What is the Electrostatic Conformal Mapping Problem?

The Electrostatic Conformal Mapping Problem is a mathematical problem that involves finding a conformal mapping between two regions with different electrical properties. This mapping allows for the calculation of the electric potential and field in one region based on the known values in the other region.

## 2. Why is the Electrostatic Conformal Mapping Problem important?

The Electrostatic Conformal Mapping Problem has many practical applications in engineering and physics, such as in the design of electrical devices and systems. It also has theoretical significance in the study of complex analysis and differential equations.

## 3. What are the challenges involved in solving the Electrostatic Conformal Mapping Problem?

One of the main challenges is finding an analytical solution for the conformal mapping, as it often involves highly complex mathematical calculations. Another challenge is dealing with boundary conditions and singularities in the regions being mapped.

## 4. How is the Electrostatic Conformal Mapping Problem typically solved?

The most common approach is to use numerical methods, such as the Schwarz-Christoffel transformation or the boundary element method, to approximate the conformal mapping. These methods involve discretizing the regions and solving the resulting equations numerically.

## 5. Are there any real-world examples of the Electrostatic Conformal Mapping Problem being used?

Yes, the Electrostatic Conformal Mapping Problem has been used in the design of microelectronic circuits, as well as in the study of electrostatic fields in conductors and insulators. It has also been applied in the analysis of fluid flow in porous media and in the study of biological systems.

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