- #1

Opus_723

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- 3

## Homework Statement

Show that the transformation

[itex]w = e^{z} + z[/itex]

maps the infinite lines [itex]y = \pm\pi[/itex] into semi-infinite lines [itex]u \leq u_{0}, v = \pm\pi[/itex]. This is equivalent to transforming an infinite or edgeless parallel-plate capacitor (z-plane) into a parallel plate capacitor (w-plane). Sketch the equipotentials in the w-plane near the edge of the capacitor plates.

Find the electric field at the plane midway between the two plates (at v = 0) as a function of u.

## Homework Equations

I found parametric equations for the equipotentials:

[itex] u = x + cos(\frac{\pi*V}{V_{0}})*e^{x}[/itex]

[itex] v = \frac{\pi*V}{V_{0}} + sin(\frac{\pi*V}{V_{0}})*e^{x}[/itex]

## The Attempt at a Solution

But I don't know how to find the electric field as a function of u along the axis. If I had an expression for V I would simply take the gradient, but buried as it is inside these parametric equations, I don't know how to get at it. I managed to elimnate x:

[itex] x = ln(\frac{v-\pi*V/V_{0}}{sin(\frac{\pi*V}{V_{0}})})[/itex]

But I'm not sure how to proceed from there.