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Condensed matter magnetism problem

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The existence of the surface of a piece of magnetizable material profoundly alters the internal magnetic field, H(int).
Mostly this is a nuisance as in the demagnetization correction required to determine the true, (i.e. internal) susceptibility of a sample.

Demagnetization can however be put to good use. One example is the magnetic shielding of instruments and experimental chambers.

Following the method used to derive the field of a uniformly magnetized sphere, derive the field in the cavity of a hollow sphere (see attached pdf).

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Problem: Consider a spherical shell made of a material with magnetic permeability  and with an inner and outer radius of a and b, respectively. This object is placed in a uniform and constant magnetic field Ha. Determine the field H inside the spherical shell (r < a). Figure 1 illustrates the geometry.

Figure 1

Solution: In this steady-state problem, no currents exist (J = 0). This means that the magnetic field H is constant and is determined from the scalar potential

H = -. (1)

As B = r0H = H and  ∙ B = 0 yields

 ∙ H =  ∙ (r0H) =  ∙ H = 0, (2)

where r (is the relative (absolute) magnetic permeability of the shell material and 0 is the permeability of free space. Combining eq. (1) and (2) gives

 ∙( ) = 2 = 0, ...

Solution Summary

The solution assists with deriving the field in the cavity of a hollow sphere.

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I am having trouble understanding this, so please answer the following questions to clarify it for me.

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