Physics Home Work - Electromagnetic

RR_UWC
timer Asked: Nov 3rd, 2018

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PHYS 4198 Homework #2 Name Problem #1 When analyzing Stoner-Wohlfarth model we got the equation form magnetization m=M/MS=cosθ in (relative to saturation) as a function of the applied field h=H/Ha (relative to the anisotropy field Ha): 2𝑚√1 − 𝑚2 cos 2𝜃0 + sin 2𝜃0 (1 − 2𝑚2 ) ± 2ℎ√1 − 𝑚2 = 0 Using this formula derive the dependence of the magnetization M(H) on the field H as it approaches to saturation. You can assume that for large fields h>>1 and m close to 1. Problem #2 Both cobalt and and Neodymium-Iron-Boron have uniaxial anisotropy. The anisotropy constant for Co is KuCo=4.5x105 J/m3 , Curie temperature TcCo=1400 K and average lattice parameter aCo=3 nm and for NdFeB the respective values are KuNdFeB=4x105 J/m3 , TcNdFeB=585 K and aNdFeB=1 nm Calculate the domain wall width and energy od domain wall per unit area for Co and NdFeB. Problem #3 Assume that the stress in magnetostrictive materials with positive magnetostriction λs varies periodically along x direction according to formula: 2𝜋𝑥 )] 𝑙 𝜎(𝑥) = 𝜎0 [1 − cos ( where σ0 is the magnitude of the stress and l is the spatial period. Assume that the area of the domain wall is A and its thickness δ. a) Derive the equation for the displacement of domain wall when field H is applied. b) Assuming the x is small so that you can make an approximation sin(2πx/l)≈2πx/l find formula for magnetic susceptibility and discuss the result. Problem #4 1 6 The magnetostatic energy density of a spherical void in ferromagnetic material is 𝐸𝑚𝑠 = 𝜇0 𝑀𝑠2 If domain wall energy per unit area is 2x10-3 J/m2 and saturation magnetization of iron is Ms=1.7x106 A/m compare the change of the magnetostatic energy with the change of the energy of the domain wall when it passes voids with radii r1=5x10-8m and r2=1x10-6m. Discuss the results. Problem #5 AMRI scientist made an array of cylindrical nano-pillars of Co with crystallographic axis c along the pillars. The height h=100 nm and diameter d=10 nm for which the demagnetizing factor is N=0.0172. The uniaxial anisotropy constant of Co is Ku=6.8x104 J/m3 and its saturation magnetization Ms=1.4x106 A/m. The linear magnetostriction constant λs=-20x106. The scientist is using a nano-manipulator to press the top of the pillar and measure the response of magnetization. What force F must he apply to deflect magnetization of the pillar by 45o from the direction normal to the substrate? Notes: In the problem of finding the relation between the magnetization and field when the field approaches saturation you can skip one of three terms in the equation of Stoner-Wohlfarth model because of conditions of large h and reduced magnetization approaching 1 In problem #2 you will need the number of nearest neighbors which is 12 for hexagonal lattice of Co and 4 for the cubic lattice of NdFeB.
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