20075ee1_1_HW4 - CHAPTER 4 4.1 The value of E at P = 2 = 40...

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CHAPTER 4 4.1. The value of E at P ( ρ = 2 , φ = 40 , z = 3) is given as E = 100 a ρ 200 a φ + 300 a z V/m. Determine the incremental work required to move a 20 µ C charge a distance of 6 µ m: a) in the direction of a ρ : The incremental work is given by dW = q E · d L , where in this case, d L = a ρ = 6 × 10 6 a ρ . Thus dW = (20 × 10 6 C)(100V / m)(6 × 10 6 m) = 12 × 10 9 J = 12nJ b) in the direction of a φ : In this case d L = 2 a φ = 6 × 10 6 a φ , and so dW = (20 × 10 6 )( 200)(6 × 10 6 ) = 2 . 4 × 10 8 J = 24 nJ c) in the direction of a z : Here, d L = dz a z = 6 × 10 6 a z , and so dW = (20 × 10 6 )(300)(6 × 10 6 ) = 3 . 6 × 10 8 J = 36 nJ d) in the direction of E : Here, d L = 6 × 10 6 a E , where a E = 100 a ρ 200 a φ + 300 a z [100 2 + 200 2 + 300 2 ] 1 / 2 = 0 . 267 a ρ 0 . 535 a φ + 0 . 802 a z Thus dW = (20 × 10 6 )[100 a ρ 200 a φ + 300 a z ] · [0 . 267 a ρ 0 . 535 a φ + 0 . 802 a z ](6 × 10 6 ) = 44 . 9nJ e) In the direction of G = 2 a x 3 a y + 4 a z : In this case, d L = 6 × 10 6 a G , where a G = 2 a x 3 a y + 4 a z [2 2 + 3 2 + 4 2 ] 1 / 2 = 0 . 371 a x 0 . 557 a y + 0 . 743 a z So now dW = (20 × 10 6 )[100 a ρ 200 a φ + 300 a z ] · [0 . 371 a x 0 . 557 a y + 0 . 743 a z ](6 × 10 6 ) = (20 × 10 6 )[37 . 1( a ρ · a x ) 55 . 7( a ρ · a y ) 74 . 2( a φ · a x ) + 111 . 4( a φ · a y ) + 222 . 9](6 × 10 6 ) where, at P , ( a ρ · a x ) = ( a φ · a y ) = cos(40 ) = 0 . 766, ( a ρ · a y ) = sin(40 ) = 0 . 643, and ( a φ · a x ) = sin(40 ) = 0 . 643. Substituting these results in dW = (20 × 10 6 )[28 . 4 35 . 8 + 47 . 7 + 85 . 3 + 222 . 9](6 × 10 6 ) = 41 . 8nJ 42
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4.2. An electric field is given as E = 10 e y (sin2 z a x + x sin2 z a y + 2 x cos2 z a z ) V/m. a) Find E at P (5 , 0 , π/ 12): Substituting this point into the given field produces E P = 10[sin( π/ 6) a x + 5sin( π/ 6) a y + 10cos( π/ 6) a z ] = 5 a x + 25 a y + 50 3 a z b) How much work is done in moving a charge of 2 nC an incremental distance of 1 mm from P in the direction of a x ? This will be dW x = q E · dL a x = 2 × 10 9 ( 5)(10 3 ) = 10 11 J = 10 pJ c) of a y ?
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