Homework6 - Homework 6 Answers 95.658 Spring 2011...

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Homework 6 Answers, 95.658 Spring 2011, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell Problem 1 For an oscillating dipole, find the time-averaged power radiated. Fill in the steps needed to get to Eq. 9.23 in Jackson. SOLUTION In the far field, the fields due to electric dipole radiation are E =− k 2 p 4  0 k × k × p e i k r − t r and B = 0 c k 2 p 4 k × p e i k r − t r We want to get an idea of where the electromagntic energy is flowing because of this radiating dipole. Energy flow, with dimensions of energy per area per time, is described by the Poynting vector S : S = E × H There are oscillations in the energy flow because there are oscillations in the fields. Of more use is the overall flow of energy with the oscillations averaged out. If the oscillation is harmonic, as it is here, we can very simply find the time-averaged energy flow by conjugating one of the fields and dividing by two and taking the real part: < S > = 1 2 ℜ E × H * Plugging in the fields above we find: < S > = 1 2 [ k 2 p 4  0 k × k × p e i k r − t r ] × 1 0 [ 0 c k 2 p 4 k × p e i k r − t r ] * < S > =− c 2 Z 0 32 2 k 4 p 2 1 r 2 [ k × k × p ] × [ k × p ] If we sketch the vectors and do a little geometry, it is easy to see that the first term in brackets is perpendicular to the second term in brackets and that their cross product will point in the - k direction. < S > = k c 2 Z 0 32 2 k 4 p 2 1 r 2 k × k × p ∣∣ k × p If we signify θ as the angle between the propagation vector k and the dipole vector p , this becomes < S > = k c 2 Z 0 32 2 k 4 p 2 1 r 2 sin 2 In the far-field, an oscillating dipole creates energy flowing radially outwards. Now note that the
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energy also spreads out radially, and so dies out according to 1/ r 2 . If we multiply this out than we will end up with a constant no matter the observation point, which is more useful. r 2 < S > = k c 2 Z 0 32 2 k 4 p 2 sin 2 Only the energy that leaves the dipole becomes radiation. If we want to know the energy radiated and not just the energy flow in general, we dot both sides with the radial vector, which is also the unit propagation vector k .
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Homework6 - Homework 6 Answers 95.658 Spring 2011...

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