Lecture24 - Therefore, the tangent mapping is a linear...

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Therefore, the tangent mapping is a linear fraction transformation, pre- ceded by an exponential mapping and followed by a clockwise rotation by 90 . With Z = e - 2 y e 2 ix , we see that the strip S := { z C | x ( - π/ 4 ,π/ 4) } in the z -plane is mapped onto the right half of the Z -plane, X> 0. The linear fractional transformation maps the right half of the Z -plane onto the interior of the unit disk in the W -plane. A clockwise rotation by 2fo l lows ,sothat w =tan z maps the strip S in the z -plane onto the interior of the unit disk in the w -plane. 18 Complex Analysis and Potential Theory The theory of solutions of Laplace’s equation (cf. Sec. 12.10) is called poten- tial theory. Laplace’s equations occurs in many Felds such as gravitation, electrostatics, heat conduction, ±uid ±ow etc. In two space dimensions, and in cartesian coordinates, we are interested in solutions of the second-order PDE ∆Φ = Φ xx yy =0 , in Ω R 2 , (1011) for the (real-valued) potential Φ:Ω R . We know from Section 13.4 that solutions of (1011) with continuous second derivatives (harmonic functions) are closely related to holomorphic functions Φ + i ΨwhereΨisaharmon ic conjugate of Φ. The restriction to two space dimensions is often justiFed when the poten- tial Φ is independent of one of the space coordinates. In this last chapter of the lecture, we shall consider this connection and its consequences in detail and illustrate it by typical boundary value problems from electrostatics, heat conduction, and hydrodynamics. 18.1 Electrostatic Fields The electric force F [ N ] between charged particles is given by Coulomb’s law. This force is the gradient of a function Φ: F = q E , E = -∇ Φ, with the electric Feld E [Vm - 1 ] = [NC - 1 ]andthe electrostatic potential Φ[V ] (cf. Sec. 12.10; there we looked at Newton’s law of universal gravitation, but Coulomb’s law has a similar form). Equipotential surfaces are level sets of Φ const . ; lines in 2D). The electric Feld is perpendicular to these surfaces. 183
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Examples: 1. Two parallel plates kept at potentials Φ 1 2 , respectively, extending to infnity. Choose a cartesian coordinate system ( x, y, z ) such that the plates are parallel to the yz -plane, and located at x = - 1and x =1 . Due to translation invariance oF the problem in both y -and z -direction, we may restrict our consideration to the x -d irect ion ,wherewehavea second-order ODE with Dirichlet boundary conditions: Φ ′′ =0 , in ( - 1 , 1) , Φ( - 1) = Φ 1 , Φ(1) = Φ 2 . (1012) The general solution is given by Φ( x )= ax + b , and the constants a, b are determined From the boundary conditions: Φ( - 1) = - a + b 1 Φ(1) = a + b 2 a = 1 2 2 - Φ 1 ) b = 1 2 1 2 ) (1013) ThereFore, the potential between two parallel plates is given by Φ( x 1 2 2 - Φ 1 ) x + 1 2 1 2 ) . (1014) The equipotential surFaces are parallel planes, x =const .
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This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.

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Lecture24 - Therefore, the tangent mapping is a linear...

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