- #1

Rlwe

- 18

- 1

- 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?