PHYS310_homework_05_solutions

PHYS310_homework_05_solutions - x y F dr F dx F dy ydx xdy...

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Homework set 5, due Wednesday, September 28, 11:30 am 1. 4.2 (20 points) (a) ˆ ˆ ˆ 0 i j k F x y z x y z ∇× = = v v conservative (b) ( 29 2 ˆ ˆ ˆ ˆ 1 1 0 i j k F k x y z y x z ∇× = = - - - v v conservative (c) ( 29 3 ˆ ˆ ˆ ˆ 1 1 0 i j k F k x y z y x z ∇× = = - = v v conservative (d) 2 ˆ ˆ ˆ sin 1 0 sin 0 0 r n e e r e r F r r kr θ φ - ∇× = = - v v conservative 2. 4.4 (20 points) (a) E = constant ( 29 2 1 , , 2 V x y z mv = + at the origin 2 1 0 2 E mv = + o at ( 29 1,1,1 2 2 1 1 2 2 E mv mv α β γ = + + + = o ( 29 2 2 2 v v m = - + + o
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( 29 1 2 2 2 v v m α β γ = - + + o (b) ( 29 2 2 0 v m - + + = o ( 29 1 2 2 v m = + + o (c) x V mx F x = = - mx = - 2 V my y y = - = - 2 3 V mz z z = - = - 3. 4.5 (20 points) (a) ˆ ˆ F ix jy = + r on the path x y = : ˆ ˆ dr idx jdy = + v ( 29 ( 29 1,1 1 1 1 1 0,0 0 0 0 0 1 x y F dr F dx F dy xdx ydy = + = + = v v on the path along the x-axis: ˆ dr idx = v and on the line 1 x = : ˆ dr jdy = v ( 29 ( 29 1,1 1 1 0,0 0 0 1 x y F dr F dx F dy = + = v v F v is conservative. (b) ˆ ˆ F iy jx = - r on the path x y = : ( 29 ( 29 1,1 1 1 1 1 0,0
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Unformatted text preview: x y F dr F dx F dy ydx xdy ⋅ = + =-∫ ∫ ∫ ∫ ∫ v v and, with x y = ( 29 ( 29 1,1 1 1 0,0 F dr xdx ydy ⋅ =-= ∫ ∫ ∫ v v on the x-axis: ( 29 ( 29 1,0 1 1 0,0 x F dr F dx ydx ⋅ = = ∫ ∫ ∫ v v and, with y = on the x-axis ( 29 ( 29 1,0 0,0 F dr ⋅ = ∫ v v on the line 1 x = : ( 29 ( 29 1,1 1 1 1,0 y F dr F dy xdy ⋅ = = ∫ ∫ ∫ v v and, with 1 x = ( 29 ( 29 1,1 1 1,0 1 F dr dy ⋅ = = ∫ ∫ v v on this path ( 29 ( 29 1,1 0,0 0 1 1 F dr ⋅ = + = ∫ v v F v is not conservative....
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This note was uploaded on 12/07/2011 for the course PHYS 310 taught by Professor Jones,m during the Fall '08 term at Purdue.

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PHYS310_homework_05_solutions - x y F dr F dx F dy ydx xdy...

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