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Capacitance of infinitely long coaxial cylinders of elliptical section
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[QUOTE="Fred Wright, post: 6581831, member: 570722"] 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}##. [/QUOTE]
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Capacitance of infinitely long coaxial cylinders of elliptical section
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