because there is zero volume magnetic charge density in a vacuum, or a uniformly magnetized magnetic medium. However, according to Equation (711), there is a magnetic surface charge density,

(718) |

on the surface of the sphere. Here, and are spherical coordinates. One of the matching conditions at the surface of the sphere is that the tangential component of must be continuous. It follows from Equation (701) that the scalar magnetic potential must be continuous at , so that

Integrating Equation (703) over a Gaussian pill-box straddling the surface of the sphere yields

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 Equation (718) that satisfies physical boundary conditions at and is

(721) |

for , and

(722) |

for . The boundary condition (720) yields

(723) |

for all . The boundary condition (721) gives

(724) |

for all , because . It follows that

(725) |

for , and

(726) | ||

(727) |

Thus,

(728) |

for , and

for . Because there is a uniqueness theorem associated with Poisson's equation (see Section 2.3), we can be sure that this axisymmetric potential is the only solution to the problem that satisfies physical boundary conditions at and infinity.

In the vacuum region outside the sphere,

(730) |

It is easily demonstrated from Equation (730) that

(731) |

where

(732) |

This, of course, is the magnetic field of a magnetic dipole of moment . [See Section 5.5.] 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

and

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

(735) |

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 Figure 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.