HW03 - x iy and w = i Problem 3.2(15 points Formulate the...

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C:\User\Teaching\505-506\F10\HW03.doc PHY 505 Classical Electrodynamics I Fall 2010 Homework 3 (due Friday Oct. 1) Problem 3.1 (to be graded of 15 points). Calculate the mutual capacitance (per unit length) between two cylindrical conductors, forming a system with the cross-section shown in Fig. on the right, in the limit t << w << R . Hint : You may like to use the “ elliptical ” (not “ellipsoidal”!) coordinates defined by the following equation: ), cosh( i c iy x (*) with the appropriate choice of constant c . In these orthogonal 2D coordinates, the Laplace operator is very simple: 2 2 2 2 2 2 2 2 ) cos (cosh 1 c . (This is not quite surprising, because Eq. (*) may be also considered as a conformal map z = c cosh
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Unformatted text preview: x + iy , and w = + i .) Problem 3.2 (15 points). Formulate the 2D electrostatic problems which can be solved using each of the following analytical functions of the complex variable z x + iy : (i) w = ln z , (ii) w = z 1/2 . Solve one (any) of these problems. Problem 3.3 (20 points). Each electrode of a large plane capacitor is cut into long strips of equal width l , with very narrow gaps between them. These strips are kept at the alternating potentials, as shown in Fig. on the right. Use the variable separation method to calculate the electrostatic potential distribution inside the capacitor. Explore the limit l << d . R w t … … … … d l 2 V 2 V 2 V 2 V 2 V 2 V...
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