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 = r0H = H and ∙ B = 0 yields
→
∙ H = ∙ (r0H) = ∙ 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, ...
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