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Unformatted text preview: CHAPTER 10 10.1. In Fig. 10.4, let B = 0 . 2cos120 πt T, and assume that the conductor joining the two ends of the resistor is perfect. It may be assumed that the magnetic field produced by I ( t ) is negligible. Find: a) V ab ( t ): Since B is constant over the loop area, the ﬂux is Φ = π (0 . 15) 2 B = 1 . 41 × 10 − 2 cos120 πt Wb. Now, emf = V ba ( t ) = − d Φ /dt = (120 π )(1 . 41 × 10 − 2 )sin120 πt . Then V ab ( t ) = − V ba ( t ) = − 5 . 33sin120 πt V . b) I ( t ) = V ba ( t ) /R = 5 . 33sin(120 πt ) / 250 = 21 . 3sin(120 πt ) mA 10.2. In Fig. 10.1, replace the voltmeter with a resistance, R . a) Find the current I that ﬂows as a result of the motion of the sliding bar: The current is found through I = 1 R I E · d L = − 1 R d Φ m dt Taking the normal to the path integral as a z , the path direction will be counterclockwise when viewed from above (in the − a z direction). The minus sign in the equation indicates that the current will therefore ﬂow clockwise , since the magnetic ﬂux is increasing with time. The ﬂux of B is Φ m = Bdvt , and so  I  = 1 R d Φ m dt = Bdv R (clockwise) b) The bar current results in a force exerted on the bar as it moves. Determine this force: F = Z Id L × B = Z d Idx a x × B a z = Z d Bdv R a x × B a z = − B 2 d 2 v R a y N c) Determine the mechanical power required to maintain a constant velocity v and show that this power is equal to the power absorbed by R . The mechanical power is P m = F v = ( Bdv ) 2 R W The electrical power is P e = I 2 R = ( Bdv ) 2 R = P m 10.3. Given H = 300 a z cos(3 × 10 8 t − y ) A/m in free space, find the emf developed in the general a φ direction about the closed path having corners at a) (0,0,0), (1,0,0), (1,1,0), and (0,1,0): The magnetic ﬂux will be: Φ = Z 1 Z 1 300 µ cos(3 × 10 8 t − y ) dx dy = 300 µ sin(3 × 10 8 t − y )  1 = 300 µ £ sin(3 × 10 8 t − 1) − sin(3 × 10 8 t ) ¤ Wb Then emf = − d Φ dt = − 300(3 × 10 8 )(4 π × 10 − 7 ) £ cos(3 × 10 8 t − 1) − cos(3 × 10 8 t ) ¤ = − 1 . 13 × 10 5 £ cos(3 × 10 8 t − 1) − cos(3 × 10 8 t ) ¤ V b) corners at (0,0,0), (2 π ,0,0), (2 π ,2 π ,0), (0,2 π ,0): In this case, the ﬂux is Φ = 2 π × 300 µ sin(3 × 10 8 t − y )  2 π = 0 The emf is therefore 0 ....
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 Fall '08
 Joshi
 Electrical Engineering, Magnetic Field, Cos, Vab, baz

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