Electromagnetic Field Theory: A Problem Solving Approach Part 71. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Radiation from Point Dipoles 675 Figure 9-3 The strength of the electric field and power density due to a z-directed point dipole as a function of angle 9 is proportional to the length of the vector from the origin to the radiation pattern. radiation pattern. These directional properties are useful in beam steering where the directions of power flow can be controlled. The total time-average power radiated by the electric dipole is found by integrating the Poynting vector over a spherical surface at any radius r J IT 2ir Sr r2sin OdOdjs 0 k 2 C ir 7dZ 2 - I i 2ir sin30d0 4ir 3 COS 0 sin2 9 2 I loir Io IM2 12 r T 2 29 676 Radiation As far as the dipole is concerned this radiated power is lost in the same way as if it were dissipated in a resistance R P k î 2R 30 where this equivalent resistance is called the radiation resistance 31 617 3 k A In free space r 0 V xo eo 120tf thé radiation resistance is 9 dl 2 Ro 80tt2 I free space A 32 These results are only true for point dipoles where dl is much less than a wavelength dl A 1 . This verifies the validity of the quasi-static approximation for geometries much smaller than a radiated wavelength as the radiated power is then negligible. If the current on a dipole is not constant but rather varies with z over the length the only term that varies with z for the vector potential in 5 is z Az r Re r dl 2 J-dl 2 fiî z e kr p ----a---------dz 4 trrQp . Re fi e jkr 1P f dl 2 - 477Tqp J-dl 2 î z dz 33 where because the dipole is of infinitesimal length the distance TqP from any point on the dipole to any field point far from the dipole is essentially r independent of z. Then all further results for the electric and magnetic fields are the same as in Section 9-2-3 if we replace the actual dipole length dl by its effective length j dl 2 dletf J- I z dz 34 A J-dl 2 where Io is the terminal current feeding the center of the dipole. Generally the current is zero at the open circuited ends as for the linear distribution shown in Figure
