# Capacitance of infinitely long coaxial cylinders of elliptical section

• Rlwe
In summary: You will have to find an expression for the potential in the ##w## plane and evaluate it at the disk boundary to find the capacitance. I can't do that for you.In summary, the conversation discusses a proof of an inequality involving special coordinates to solve a system of confocal ellipses. The speaker is unsure of how to obtain the exact value and asks for assistance. They also mention a paper on a similar topic. The other person suggests using the Joukowski conformal transformation to map the ellipses to disks and gives equations for determining the capacitance in the transformed plane.
Rlwe
Homework Statement
Find the capacitance per unit length between two infinitely long coaxial cylinders of elliptical section given by eqs. $$\frac{x^2}{a_1^2}+\frac{y^2}{b_1^2}=1$$ $$\frac{x^2}{a_2^2}+\frac{y^2}{b_2^2}=1$$ where $$\frac{a_2}{a_1}=\frac{b_2}{b_1}$$ and $$b_1\geq a_1\,,\quad b_2\geq a_2\,,\quad a_2>a_1$$
Relevant Equations
Laplace equation in 2D
I've been able to prove the following inequality $$\frac{2\pi\epsilon_0}{\log\left(\frac{b_1b_2}{a_1^2}\right)}\leq C \leq \frac{2\pi\epsilon_0}{\log\left(\frac{a_1a_2}{b_1^2}\right)}$$ but have no clue how to obtain exact value. Can someone check whether this inequality is correct and show how to obtain the exact value?

vela said:
Sorry, it isn't really a homework (maybe I shouldn't have posted it under HW help, sorry) but a problem I invented. I found this paper (p.10) which deals with a system of confocal ellipses and uses special system of coords to solve it. However, I couldn't find any reference that deals with similar ellipses.

Rlwe said:
it isn't really a homework (maybe I shouldn't have posted it under HW help, sorry) but a problem I invented.
It is still schoolwork-like, so you did the right thing to post it in the schoolwork forums.

I suggest you use the Joukowski conformal transformation to map ellipses in the ##z## plane (##z=x+iy##) to disks in the ##w## plane (##w=u+iv##), then compute the capacitance in the ##w## plane with cylindrical symmetry (not hard). The Joukowski transformation is,
$$z=\alpha w + \frac{\beta}{w}$$
$$z_{1,2}=x_{1,2}+iy_{1,2}=\alpha_{1,2} (u_{1,2}+iv_{1,2}) + \beta_{1,2} \frac{u_{1,2}-iv_{1,2}}{R_{1,2}^2}$$
where ##R_{1,2}## are the radii of the disks in the ##w## plane. Equating real and imaginary parts, the equation ## u^2_{1,2} + v^2_{1,2}=R^2_{1,2}## becomes,
$$\frac{x^2_{1,2} }{\alpha_{1,2} + \frac{\beta_{1,2} }{R^2_{1,2}}} +\frac{y^2_{1,2} }{\alpha_{1,2} - \frac{\beta_{1,2} }{R^2_{1,2}}}=1$$
where,
$$a_{1,2}=|\alpha_{1,2} + \frac{\beta_{1,2} }{R^2_{1,2}} |$$
$$b_{1,2}=|\alpha_{1,2} - \frac{\beta_{1,2} }{R^2_{1,2}} |$$
You will have to choose a scale for your problem i.e. ##a_2=\gamma a_1## and ##b_2=\gamma b_1##. From this and the given conditions and constraints you can compute the ratio ##\frac{R_2}{R_1}##.

vanhees71 and berkeman

## 1. What is the formula for calculating the capacitance of infinitely long coaxial cylinders of elliptical section?

The formula for calculating the capacitance of infinitely long coaxial cylinders of elliptical section is C = (2πε0εr)/(ln(b/a)), where a and b are the major and minor axis of the ellipse, ε0 is the permittivity of free space, and εr is the relative permittivity of the material between the cylinders.

## 2. How does the capacitance of infinitely long coaxial cylinders of elliptical section differ from that of circular cylinders?

The capacitance of infinitely long coaxial cylinders of elliptical section is larger than that of circular cylinders with the same radius, due to the increased surface area of the elliptical cross-section. This results in a larger electric field and therefore a larger capacitance.

## 3. Can the capacitance of infinitely long coaxial cylinders of elliptical section be increased?

Yes, the capacitance of infinitely long coaxial cylinders of elliptical section can be increased by decreasing the distance between the cylinders, increasing the relative permittivity of the material between the cylinders, or increasing the length of the cylinders.

## 4. How does the shape of the elliptical cross-section affect the capacitance of infinitely long coaxial cylinders?

The capacitance of infinitely long coaxial cylinders is directly proportional to the logarithm of the ratio of the major and minor axis of the ellipse. This means that as the ellipse becomes more elongated, the capacitance increases.

## 5. What is the significance of the capacitance of infinitely long coaxial cylinders of elliptical section?

The capacitance of infinitely long coaxial cylinders of elliptical section is an important factor in determining the performance of electrical systems, such as transmission lines and coaxial cables. It also plays a role in the design and operation of electronic devices and circuits.

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