ps12 - 5. Consider two non-concentric circles, one of which...

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MATH 406: Problem Set 12 due Friday, April 22, 2005 (Revised version) 1. Find the roots of the following polynomials; show your work, recommended not to use a calculator! a) z 2 + (2 i - 7) z - 14 i b) z 2 + ( i + 8) z + 8 i c) z 2 - (12 i + 9) z + 108 i 2. Fisher 4.2.1 3. Fisher 4.2.2 4. Find the electrostatic potential φ between two infinite flat plates in the plane: one along the positive real axis, at φ 0 = 0, and one starting at the origin and situated up and to the left, meeting the first plate at 120 , held at φ 1 . Note that there is a small insulator between the two plates at the origin!
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Unformatted text preview: 5. Consider two non-concentric circles, one of which crosses the axis at x = 1 / 7 and x = 1 / 2 (with its center on the real axis), and one dened by | z 2 | = 1. First show that the following map w = 3 z-1 3-z a) moves the inner circle to | w 1 | = 1 / 5, and b) keeps the outer circle xed ( | w 2 | = 1). Next nd the electrostatic potential between the two original cylinders in the z-plane, with the inner one held xed at 1 = 12 V and the outer one held xed at 2 = 40 V. Version 1.2...
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