Mathematics 317 Solutions to Assignment6(2)

Mathematics 317 - Mathematics 317 Solutions to Assignment#6 1 F = Hence a line of force lies in the direction of But equipotential surface = const

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Mathematics 317 Solutions to Assignment #6 1. . ϕ = F Hence a line of force lies in the direction of . But equipotential surface = const. Hence lines of force are perpendicular to equipotential surfaces. 2. (a) Clearly any vector field of the form k j i F ) ( ) ( ) ( z c y b x a + + = is conservative with potential . ) ( ) ( ) ( ) , , ( + + = dz z c dy y b dx x a z y x For the given example . ) ( , ) ( , ) ( 2 2 2 z z c y y b x x a = = = Hence ). ( ) , , ( 3 3 3 3 1 z y x z y x + + = (b) Here k j i F ) , , ( ) , , ( ) , , ( ) , , ( 3 2 1 z y x F z y x F z y x F z y x + + = where . ) , , ( , ) , , ( , ) , , ( 3 2 1 xy z y x F x z y x F y z y x F = = = If F = , then . , , 3 2 1 xy F x F y F z y x = = = = = = For a potential ) , , ( z y x to exist, it is necessary that , zy yz = i.e., . 3 2 y F z F = In this example, . , 0 2 3 2 z F x y F z F = = Hence F is not conservative. 3.
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This note was uploaded on 04/01/2010 for the course MATH 317 MATH 317 taught by Professor Bluman during the Spring '10 term at The University of British Columbia.

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