Capacitance of infinitely long coaxial cylinders of elliptical section

[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]

Please show us your work.

 

[Rlwe]

Please show us your work.

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.

 

[berkeman]

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.

 

[Gordianus]

This article may help (I haven't read it in depth)
10.2528/PIER03072303

 

[Fred Wright]

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 $\frac{R_2}{R_1}$.