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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 pillbox 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
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.
Figure 4:
Schematic demagnetization curve for a permanent magnet

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
20020518