tailieunhanh - Electromagnetic Field Theory: A Problem Solving Approach Part 38

Electromagnetic Field Theory: A Problem Solving Approach Part 38. 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. | Magnetization 345 Using the law of cosines these distances are related as 2 where the angles Xi and Xz are related to the spherical coordinates from Table 1-2 as ir i cos Xi sin 0 sin d ir ix cos X2 sin 6 cos t In the far field limit 1 becomes 1 lim A r dx r dy Tl 2 cosxi I 1 11 2 cosxi 1 l L 2r dx 1 2 - 2 cos 2 2r J - rdy cos Xiix cos y2ij 4 4ttt Using 3 4 further reduces to A dS 4t7t2 sin 0 sin ix cos iy u0I dS ----2 sin 01 5 where we again used Table 1-2 to write the bracketed Cartesian unit vector term as i . The magnetic dipole moment m is defined as the vector in the direction perpendicular to the loop in this case iz by the right-hand rule with magnitude equal to the product of the current and loop area m I dS iz I dS 6 346 The Magnetic Field Then the vector potential can be more generally written as fiom . xom . A - 2 sin 01 - j x r 7 4ttt 47rr with associated magnetic field Id 13 B V x A A sin 0 ir rA i r sin 0 30 r dr uotn n - 3 2 cos flir sin 0i 8 4ttt This field is identical in form to the electric dipole field of Section 3-1-1 if we replace p E0 by i07n. 5-5-2 Magnetization Currents Ampere modeled magnetic materials as having the volume filled with such infinitesimal circulating current loops with number density N as illustrated in Figure 5-15. The magnetization vector M is then defined as the magnetic dipole density M Am NI dS amp m 9 For the differential sized contour in the xy plane shown in Figure 5-15 only those dipoles with moments in the x or y directions thus z components of currents will give rise to currents crossing perpendicularly through the surface bounded by the contour. Those dipoles completely within the contour give no net current as the current passes through the contour twice once in the positive z direction and on its return in the negative z direction. Only those dipoles on either side of the edges so that the current only passes through the contour once with the return outside the contour give a net current through the loop.