Capacitor Fringe Field Strength on Axis

[Opus_723]

Homework Statement

Show that the transformation

$w = e^{z} + z$

maps the infinite lines $y = \pm\pi$ into semi-infinite lines $u \leq u_{0}, v = \pm\pi$. 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:

$u = x + cos(\frac{\pi*V}{V_{0}})*e^{x}$
$v = \frac{\pi*V}{V_{0}} + sin(\frac{\pi*V}{V_{0}})*e^{x}$

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:

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

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

 

[Opus_723]

Any help would be greatly appreciated. I'm stuck. However, I'm convinced that my parametric equations are correct, so its just a matter of how to go to the electric field from there. Since V is constant on the u-plane, I only need to take the partial derivative of V with respect to the vertical coordinate v (unfortunate choice of letters, I should have changed that). But I don't know how to take a partial derivative when the potential V is buried inside these parametric equations.