Integration with variables Notes_Part_15

Integration with variables Notes_Part_15 - Figure 7.25. A...

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Unformatted text preview: Figure 7.25. A NonCoaxial Cable. Example 7.39. A non-coaxial cable . The goal of this example is to determine the electrostatic potential inside a non-coaxial cylindrical cable, as illustrated in Figure 7.25, with prescribed constant potential values on the two bounding cylinders. Assume for definiteness that the larger cylinder has radius 1, and is centered at the origin, while the smaller cylinder has radius 2 5 , and is centered at z = 2 5 . The resulting electrostatic potential will be independent of the longitudinal coordinate, and so can be viewed as a planar potential in the annular domain contained between two circles representing the cross-sections of our cylinders. The desired potential must satisfy the Dirichlet boundary value problem u = 0 when | z | < 1 and vextendsingle vextendsingle z 2 5 vextendsingle vextendsingle > 2 5 , u = a, when | z | = 1 , and u = b when vextendsingle vextendsingle z 2 5 vextendsingle vextendsingle = 2 5 . According to Example 7.36, the linear fractional transformation = 2 z 1 z 2 (7 . 92) maps this non-concentric annular domain to the annulus A 1 / 2 , 1 = braceleftbig 1 2 < | | < 1 bracerightbig , which is the cross-section of a coaxial cable. The corresponding transformed potential U ( , ) has the constant Dirichlet boundary conditions U = a, when | | = 1 2 , and U = b when | | = 1 . (7 . 93) Clearly the coaxial potential U must be a radially symmetric solution to the Laplace equation, and hence, according to (6.103), of the formequation, and hence, according to (6....
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This note was uploaded on 02/10/2012 for the course MATH 5587 taught by Professor Olver during the Fall '10 term at University of Central Florida.

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Integration with variables Notes_Part_15 - Figure 7.25. A...

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