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## Boundary value problems with ferromagnets

Consider a ferromagnetic sphere, of uniform permeability , placed in a uniform -directed magnetic field of magnitude . Suppose that the sphere is centred on the origin. In the absence of any true currents, we have . Hence, we can write . Given that , and , it follows that in any uniform magnetic medium (or a vacuum). Hence, throughout space. Adopting spherical polar coordinates, , aligned along the -axis, the boundary conditions are that at , and that is well-behaved at . At the surface of the sphere, , the continuity of implies that is continuous. Furthermore, the continuity of leads to the matching condition
 (875)

Let us try separable solutions of the form . It is easily demonstrated that such solutions satisfy Laplace's equation provided that or . Hence, the most general solution to Laplace's equation outside the sphere, which satisfies the boundary condition at , is

 (876)

Likewise, the most general solution inside the sphere, which satisfies the boundary condition at , is
 (877)

The continuity of at yields
 (878)

Likewise, the matching condition (875) gives
 (879)

Hence,
 (880) (881)

Note that the magnetic field inside the sphere is uniform, parallel to the external magnetic field outside the sphere, and of magnitude . Moreover, , provided that .

As a final example, consider an electromagnet of the form sketched in Fig. 50. A wire, carrying a current , is wrapped times around a thin toroidal iron core of radius and permeability . The core contains a thin gap of width . What is the magnetic field induced in the gap? Let us neglect any leakage of magnetic field from the core, which is reasonable if . We expect the magnetic field, , and the magnetic intensity, , in the core to be both toroidal and essentially uniform. It is also reasonable to suppose that the magnetic field, , and the magnetic intensity, , in the gap are toroidal and uniform, since . We have and . Moreover, since the magnetic field is normal to the interface between the core and the gap, the continuity of implies that
 (882)

Thus, the magnetic field-strength in the core is the same as that in the gap. However, the magnetic intensities in the core and the gap are quite different: . Integration of Eq. (871) around the torus yields
 (883)

Hence,
 (884)

It follows that
 (885)

Next: Magnetic energy Up: Dielectric and magnetic media Previous: Boundary conditions for and
Richard Fitzpatrick 2006-02-02