# Capacitor Fringe Field Strength on Axis

• Opus_723
In summary, the transformation w = e^{z} + z maps the infinite lines y = \pm\pi into semi-infinite lines u \leq u_{0}, v = \pm\pi, which is equivalent to transforming an infinite or edgeless parallel-plate capacitor (z-plane) into a parallel plate capacitor (w-plane). The equipotentials in the w-plane near the edge of the capacitor plates can be represented by the parametric equations u = x + cos(\frac{\pi*V}{V_{0}})*e^{x} and v = \frac{\pi*V}{V_{0}} + sin(\frac{\pi*V}{V_{0}})*e^{x}. To find the

## 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.

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.

## 1. What is a capacitor fringe field?

A capacitor fringe field is the electric field that extends beyond the edges of a capacitor's plates. It is strongest at the edges and decreases in strength further away from the plates.

## 2. How is the fringe field strength on axis calculated?

The fringe field strength on axis can be calculated using the formula E = Q/(2πε₀rL), where E is the electric field strength, Q is the charge on the capacitor plates, ε₀ is the permittivity of free space, r is the distance from the center of the plates, and L is the length of the capacitor's plates.

## 3. How does the distance between the plates affect the fringe field strength on axis?

The distance between the plates has a direct effect on the fringe field strength on axis. As the distance increases, the electric field strength decreases. This is because the electric field lines from each plate have to travel a greater distance to reach the other plate, resulting in a weaker field.

## 4. What factors can affect the fringe field strength on axis?

The fringe field strength on axis can be affected by several factors, including the distance between the plates, the charge on the plates, and the length of the plates. Other factors such as the dielectric material between the plates and any external electric or magnetic fields can also have an impact.

## 5. Why is the fringe field strength on axis important?

The fringe field strength on axis is important because it can affect the performance of a capacitor. It can cause unwanted interference or distortions in electronic circuits and can also lead to energy loss in the form of heat. It is important to consider and minimize the fringe field strength when designing and using capacitors in various applications.