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chap29 - MAXWELLS EQUATIONS AND ELECTROMAGNETIC WAVES 29...

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MAXWELL’S EQUATIONS AND ELECTROMAGNETIC WAVES 29 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 EA V d A I t E 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 G G . E B × E VALUATE When E G is parallel to and ˆ j B G is parallel to the direction of propagation is parallel to ˆ , i , E B × G G or 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 which is ( E i j + ), 2. E Note that ˆ 2 , i j n + = where is a unit vector 45 ° between the positive x and y axes. (b) When ˆ n E G is parallel to (for ˆ n sin( ) kz t ω positive) points 45 ° into the second quadrant (so that Thus, B G , and is in the direction). E B E B z × + G G G G B G is parallel to the unit vector ( ) / 2. i j − + 0 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 1 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, can be rewritten as SE R 11 10 1c-min (1.5 10 m) 8.33 c-min 1.8 10 m SE R = × = × 29.1
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