The linear relation between and is expressed as 858 Where is a dimensionless

The linear relation between and is expressed as 858

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⃗⃗ . The linear relation between ? ⃗⃗ and ? 0 is expressed as ? ⃗⃗ = ? ? ? 0 ? 0 (8.5.8) Where ? ? is a dimensionless quantity called the magnetic susceptibility . The equation of the magnetic field can then be written as ? = (1 + ? ? )? 0 = ? ? ? 0 (8.5.9) Where ? ? = 1 + ? ? (8.5.10) is called the relative permeability of the material. For paramagnetic substances, m 1, or equivalently, ? ? > 0 , although ? ? is usually on the order of 10 -6 to 10 3 . The magnetic permeability ? ? of a material may also be defined as m (1 m ) 0 m 0 (8.5.11) Paramagnetic materials have m 0 .
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96 8.5.3. Diamagnetism In the case of magnetic materials where there are no permanent magnetic dipoles, the presence of an external field ? 0 will induce magnetic dipole moments in the atoms or molecules. However, these induced magnetic dipoles are anti-parallel to ? 0 , leading to a magnetization ? ⃗⃗ and average field ? M anti-parallel to ? 0 , and therefore a reduction in the total magnetic field strength. For diamagnetic materials, we can still define the magnetic permeability, as in equation (8-5), although now m 1, or m 0, although is usually on the order of 10 5 to 10 9 . Diamagnetic materials have . 8.5.4. Ferromagnetism In ferromagnetic materials, there is a strong interaction between neighboring atomic dipole moments. Ferromagnetic materials are made up of small patches called domains , as illustrated in Figure 8.5.3(a). An externally applied field ? 0 will tend to line up those magnetic dipoles parallel to the external field, as shown in Figure 8.5.3(b). The strong interaction between neighboring atomic dipole moments causes a much stronger alignment of the magnetic dipoles than in paramagnetic materials. Figure 8.5.3 (a) Ferromagnetic domains. (b) Alignment of magnetic moments in the direction of the external field ? 0 . The enhancement of the applied external field can be considerable, with the total magnetic field inside a ferromagnet 10 3 or 10 4 times greater than the applied field. The permeability ? ? of a ferromagnetic material is not a constant, since neither the total field ? or the magnetization ? ⃗⃗ increases linearly with ? 0 . In fact the relationship between ? ⃗⃗ and ? 0 is not unique, but dependent on the previous history of the material. The phenomenon is known as hysteresis . The variation of ? ⃗⃗ as a function of the externally
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97 applied field ? 0 is shown in Figure 8.5.4. The loop abcdef is a hysteresis curve . Figure 8.5.4 A hysteresis curve. Moreover, in ferromagnets, the strong interaction between neighboring atomic dipole moments an keep those dipole moments aligned, even when the external magnet field is reduced to zero. And these aligned dipoles can thus produce a strong magnetic field, all by themselves, without the necessity of an external magnetic field. This is the origin of permanent magnets. To see how strong such magnets can be, consider the fact that magnetic dipole moments of atoms typically have magnitudes of the order of 10 23 A m 2 . Typical atomic densities are 10 29 atoms/m 3 . If all these dipole moments are aligned, then we would get a magnetization of order M (10 23 A m 2 )(10 29 atoms/m 3 ) 10 6 A/m (8.5.12)
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