# 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?

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Homework Helper

Rlwe
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.

Mentor
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
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}##.

vanhees71 and berkeman