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A uniformly magnetized sphere

Consider a sphere of radius , with a uniform permanent magnetization , surrounded by a vacuum region. The simplest way of solving this problem is in terms of the scalar magnetic potential introduced in Eq. (3.134). From Eqs. (3.136) and (3.137), it is clear that satisfies Laplace's equation,
 (572)

since there is zero volume magnetic charge density in a vacuum or a uniformly magnetized magnetic medium. However, according to Eq. (3.144), there is a magnetic surface charge density,
 (573)

on the surface of the sphere. One of the matching conditions at the surface of the sphere is that the tangential component of must be continuous. It follows from Eq. (3.134) that the scalar magnetic potential must be continuous at , so that
 (574)

Integrating Eq. (3.136) over a Gaussian pill-box straddling the surface of the sphere yields
 (575)

In other words, the magnetic charge sheet on the surface of the sphere gives rise to a discontinuity in the radial gradient of the magnetic scalar potential at .

The most general axisymmetric solution to Eq. (3.151) which satisfies physical boundary conditions at and is

 (576)

for , and
 (577)

for . The boundary condition (3.153) yields
 (578)

for all . The boundary condition (3.154) gives
 (579)

for all , since . It follows that
 (580)

for , and
 (581) (582)

Thus,
 (583)

for , and
 (584)

for . Since there is a uniqueness theorem associated with Poisson's equation, we can be sure that this axisymmetric potential is the only solution to the problem which satisfies physical boundary conditions at and infinity.

In the vacuum region outside the sphere

 (585)

It is easily demonstrated from Eq. (3.162) that
 (586)

where
 (587)

This, of course, is the magnetic field of a magnetic dipole . Not surprisingly, the net dipole moment of the sphere is equal to the integral of the magnetization (which is the dipole moment per unit volume) over the volume of the sphere.

Inside the sphere we have and , giving

 (588)

and
 (589)

Thus, both the and fields are uniform inside the sphere. Note that the magnetic intensity is oppositely directed to the magnetization. In other words, the field acts to demagnetize the sphere. How successful it is at achieving this depends on the shape of the hysteresis curve in the negative and positive quadrant. This curve is sometimes called the demagnetization curve of the magnetic material which makes up the sphere. Figure 4 shows a schematic demagnetization curve. The curve is characterized by two quantities: the retentivity (i.e., the residual magnetic field strength at zero magnetic intensity) and the coercivity (i.e., the negative magnetic intensity required to demagnetize the material: this quantity is conventionally multiplied by to give it the units of magnetic field strength). The operating point (i.e., the values of and inside the sphere) is obtained from the intersection of the demagnetization curve and the curve . It is clear from Eqs. (3.166) and (3.167) that
 (590)

for a uniformly magnetized sphere in the absence of external fields. The magnetization inside the sphere is easily calculated once the operating point has been determined. In fact, . It is clear from Fig. 4 that for a magnetic material to be a good permanent magnet it must possess both a large retentivity and a large coercivity. A material with a large retentivity but a small coercivity is unable to retain a significant magnetization in the absence of a strong external magnetizing field.

Next: A soft iron sphere Up: The effect of dielectric Previous: Permanent ferromagnets
Richard Fitzpatrick 2002-05-18