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CU-Boulder PHYS 3320 - Homework #12

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1 of 2 PHYS3320 Fall 2009 Phys 3320, HW #12, Due at start of class, Friday December 11th Q1. CLICKER USE ONLINE SURVEY You will get full credit (5 pts) on this homework problem just for filling out this survey online survey about the course: http://www.colorado.edu/sei/surveys/Fall09/Clicker_Phys3320_fa09-post.html . We won't grade you in any way on your specific responses. Your opinions matter and will help us improve this course in the future. The course instructor will only see anonymous results; he will not see names associated with responses. Q2. EM WAVES FROM AN OSCILLATING ELECTRIC DIPOLE In class we found that the leading terms in the radiation “multipole” expansion for the electric and magnetic fields were E(r,t) =μ04r[ˆ r (ˆ r ˙ ˙ p (t0))]B(r,t) = μ04rc[ˆ r ˙ ˙ p (t0)] A. Consider the case of two point charges, +q and –q located on the z-axis, symmetrically about the origin. The separation of the charges is given by acos(t). Find the resulting E and B fields in spherical coordinates. B. Show that the fields that you found satisfy Maxwell’s equations in vacuum, and find the time-averaged Poynting vector <S>. Make a polar plot of the magnitude of the time-averaged Poynting vector as a function of azimuthal angle  for  = /2 (r = constant in polar plots) as well as a polar plot of the magnitude as a function of  for  = 0. C. These fields represent spherical waves. Far from the origin, they can be approximated at planes waves (just as the earth’s surface appears flat to us). What is the dispersion relation for these waves? Rewrite your expressions for E and B in terms of our standard complex notation. Q3. A SMALL ANTENNA ARRAY Two sinusoidally oscillating dipoles are located in the x-y plane on the y-axis at y = a and y = -a. Their dipole moments are oriented in the z direction. Assume that our observation point r is located at a distance far away, r >> a. Also assume the dipole moments have equal amplitude p0. A. Find the total electric field E in the x-y plane using the standard complex notation for the fields. B. Assume the dipoles are a half-wavelength apart ( = 4a). Make a polar plot of |E| as a function of azimuthal angle  for the case that the oscillations of the two dipoles are in phase. C. For the same wavelength, now make a polar plot of |E| as a function of azimuthal angle  for the case that the oscillations of the two dipoles are 180° out of phase. D. Compare your polar plots with that of a single dipole located at the origin (and oriented in the z direction) of twice the amplitude, i.e., 2p0. What is the advantage of the two-dipole array over the single dipole setup?2 of 2 PHYS3320 Fall 2009 Q4. LORENTZ TRANSFORMATION FOR GENERAL VELOCITY VECTOR Textbooks tend to only give you the Lorentz transformation along a single coordinate axis, but it is not always convenient to keep redefining the coordinate system for problems with several different velocities. To derive a more general formula using vector notation, use the idea that the part of a position vector r that is parallel to the velocity is the part that is changed by the transformation, while the part that is perpendicular to the velocity is unchanged. Assume that you wish to transform from your inertial frame (the (r, ct) frame) to the “primed” inertial frame (r, ct) moving with velocity v =c which points in some arbitrary direction (e.g., it has an x, y and z component). You should find the following: c t = (ct  r ) r = r + ( 1)(r ˆ  )ˆ   ct Show that in the case that the velocity is in the x direction, you get back the regular transformation given in


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