Mathematics 317 Solutions to Assignment6(2) - Mathematics 317 Solutions to Assignment#6 1 F = Hence a line of force lies in the direction of But

# Mathematics 317 Solutions to Assignment6(2) - Mathematics...

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Mathematics 317 Solutions to Assignment #6 1. . ϕ = F Hence a line of force lies in the direction of . ϕ But ϕ equipotential surface ϕ = 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 ϕ 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 + + = ϕ (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 = ϕ , then . , , 3 2 1 xy F x F y F z y x = = = = = = ϕ ϕ ϕ For a potential ) , , ( z y x ϕ to exist, it is necessary that , zy yz ϕ ϕ = 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. F is not conservative since . 1 , 0 2 3 2 z F y F z F = = Note that j i G x y = is a conservative vector field. On any curve C for which z = const (i.e, for any curve lying on a plane parallel to the xy -plane), one has . 0 = = C C d d r G r F 4. As discussed in class, the force acting on the particle with charge Q is given by Coulomb’s Law: . 3 r F r qQ = As shown in class, - = r qQ F (and hence is