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29_InstSolManual_PC - MAXWELL'S EQUATIONS AND...

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MAXWELL’S EQUATIONS AND ELECTROMAGNETIC WAVES E XERCISES Section 29.2 Ambiguity in Ampère’s Law 13. I NTERPRET In this problem we are asked to find the displacement current through a surface. D EVELOP As shown in Equation 29.1, Maxwell’s displacement current is 0 0 0 ( ) E d d d EA dE I A dt dt dt ε ε ε Φ = = = E VALUATE The above equation gives 2 12 2 2 0 (8.85 10 C /N m )(1 cm )(1.5 V/m s) 1.33 nA d dE I A dt ε μ - = = × = A SSESS Displacement current arises from changing electric flux and has units of amperes (A), just like ordinary current. 14. The electric field is approximately uniform in the capacitor, so 0 ( / ) , and / E D E EA V d A I t φ ε φ = = = ∂ = 12 2 0 ( / ) / (8.85 10 F/m)(10 cm) (220 V/ms)/(0.5 cm) 3.89 A. A d dV dt ε μ - = × = Section 29.4 Electromagnetic Waves 15. I NTERPRET We are given the electric and magnetic fields of an electromagnetic wave and asked to find the direction of propagation. D EVELOP The direction of propagation of the electromagnetic wave is the same as the direction of the cross product . E B × r r E VALUATE When E r is parallel to ˆ j and B r is parallel to ˆ , i the direction of propagation is parallel to , E B × r r or ˆ ˆ ˆ j i k × . = - A SSESS For electromagnetic waves in vacuum, the directions of the electric and magnetic fields, and of wave propagation, form a right-handed coordinate system. 16. (a) The peak amplitude is the magnitude of ˆ ˆ ( ), E i j + which is 2. E Note that ˆ ˆ ˆ 2 , i j n + = where ˆ n is a unit vector 45 ° between the positive x and y axes. (b) When E r is parallel to ˆ n (for sin( ) kz t ϖ - positive) B r points 45 ° into the second quadrant (so that , and is in the direction). E B E B z × + r r r r Thus, B r is parallel to the unit vector ˆ ˆ ( )/ 2. i j - + Section 29.5 Properties of Electromagnetic Waves 17. I NTERPRET This problem is about measuring the distance between the Sun and the Earth using light-minutes. D EVELOP A light-minute (abbreviated as c-min) is approximately equal to 8 10 1 c-min (3 10 m/s)(60 s) 1.8 10 m = × = × On the other hand, the mean distance of the Earth from the Sun (an Astronomical Unit) is about 11 1.5 10 m. SE R = × E VALUATE In units of c-min, SE R can be rewritten as 11 10 1c-min (1.5 10 m) 8.33 c-min 1.8 10 m SE R = × = × 29.1 29
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29.2 Chapter 29 A SSESS The result implies that it takes about 8.33 minutes for the sunlight to reach the Earth. 18. Assuming the satellite is approximately overhead, we can estimate the round-trip travel time by / t r c ∆ = ∆ = 5 (2 36,000 km)/(3 10 km/s) 0.24 s. × × = 19. I NTERPRET In this problem we want to deduce the airplane’s altitude by measuring the travel time of a radio wave signal it sends out. D EVELOP The speed of light is 8 3 10 m/s c = × and the total distance traveled is 2 . r h = E VALUATE Since 2 r h c t = = (for waves traveling with speed c ), the altitude is 8 (3 10 m/s)(50 s) 7.5 km 2 2 c t h μ × = = = A SSESS The airplane is flying lower than the typical cruising altitude of 12,000 m (35,000 ft) for commercial jet airplanes.
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