Explanation Given : w = 1 z Concept used: Imaginary part of the equation is taken as temperature function. In Iso thermal process the temperature is constant. Calculation: Consider the figure: The above figure represents the hot cylinder ( T=100° ) lying on a cold plane ( T=0° ) The objective is to determine the temperature in the blue shaded region. Consider the mapping function w = 1 z , Under the mapping w = 1 z , the above diagram in w- plane is show below. Separate the real and imaginary parts of the function w. w = 1 z u + i v = 1 x + i y Rationalizing the above equation: u + i v = 1 x + i y . x − i y x − i y u + i v = x − i y x 2 + y 2 . u + i v = x x 2 + y 2 + i ( − y x 2 + y 2 ) So here the real part is u = x x 2 + y 2 and imaginary part is v = ( − y x 2 + y 2 ) . From elementary physics, the temperature decreases linearly from the top edge ( v = 0 ) to the bottom edge ( v = − 1 ) . Therefore, the temperature is T ( x , y ) = 100 y x 2 + y 2

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Chapter 14.10, Problem 6P

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Tofind:the temperature and electric potential in shaded region and sketch some isothermal or equipotential.

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Mathematical Methods in the Physical Sciences

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Tofind: the temperature and electric potential in shaded region and sketch some isothermal or equipotential.

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Tofind:the temperature and electric potential in shaded region and sketch some isothermal or equipotential.

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Chapter 14 Solutions