Next: Magnetic energy
Up: Dielectric and magnetic media
Previous: Boundary conditions for and
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 wellbehaved 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,
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 .
Figure 50:

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 fieldstrength 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
20060202