tailieunhanh - Electromagnetic Field Theory: A Problem Solving Approach Part 39

Electromagnetic Field Theory: A Problem Solving Approach Part 39. 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 355 so that the total magnetization due to all the dipoles within the sphere is . -_ WlflTV f đ CO 0 JZJ tAQX M - . I sin Q cos 0 e do 4a 2 sinh a J o Again using the change of variable in 44 48 integrates to -mN r . M - I ue du 2a sinh a Ja mN u 1x1 a o g -1 1 a 2a sinh a mN _ -a -1 - a 1 2a sinh a mN r . a cosh a sinh a a sinh a mN co th a - 1 a 49 which is known as the Langevin equation and is plotted as a function of reciprocal temperature in Figure 5-19. At low temperatures high a the magnetization saturates at M mN as all the dipoles have their moments aligned with the field. At room temperature a is typically very small. Using the parameters in 26 and 27 in a strong magnetic field of Ho 106amps m a is much less than unity a m 0 Ä2 Ä _4 kT 2 kT 50 Figure 5-19 The Langevin equation describes the net magnetization. At low temperatures high a all the dipoles align with the field causing saturation. At high temperatures a 1 the magnetization increases linearly with field. 356 The Magnetic Field In this limit Langevin s equation simplifies to r w J1 fl2 2 M hrn M m VI--r I a i La a 6 aJ l a Xl-a3 r a m. a mNa fjLOm2N s 3kTH 51 In this limit the magnetic susceptibility Xm is positive M Xm H Xm 52 OK 1 but even with N IO30 atoms m3 it is still very small Xm 7xl04 53 c Ferromagnetism As for ferroelectrics see Section 3-1-5 sufficiently high coupling between adjacent magnetic dipoles in some iron alloys causes them to spontaneously align even in the absence of an applied magnetic field. Each of these microscopic domains act like a permanent magnet but they are randomly distributed throughout the material so that the macroscopic magnetization is zero. When a magnetic field is applied the dipoles tend to align with the field so that domains with a magnetization along the field grow at the expense of nonaligned domains. The friction-like behavior of domain wall motion is a lossy process so that the magnetization varies with the magnetic field in a .