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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
![\begin{displaymath}
\left.\mu_0 \frac{\partial \phi_m}{\partial r}\right\vert _...
...ft.\mu
\frac{\partial\phi_m}{\partial r}\right\vert _{r=a-}.
\end{displaymath}](img1821.png) |
(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
![\begin{displaymath}
\phi_m(r,\theta) = - (B_0/\mu_0) r \cos\theta + (B_0/\mu_0) \alpha \frac{a^3 \cos\theta}{r^2}.
\end{displaymath}](img1822.png) |
(876) |
Likewise, the most general solution inside the sphere, which satisfies
the boundary condition at
, is
![\begin{displaymath}
\phi_m(r,\theta) = - (B_1/\mu) r \cos\theta.
\end{displaymath}](img1823.png) |
(877) |
The continuity of
at
yields
![\begin{displaymath}
B_0 - B_0 \alpha = (\mu_0/\mu) B_1.
\end{displaymath}](img1824.png) |
(878) |
Likewise, the matching condition (875) gives
![\begin{displaymath}
B_0 + 2 B_0 \alpha = B_1.
\end{displaymath}](img1825.png) |
(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:
![\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{mc.eps}}
\end{figure}](img1832.png) |
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
![\begin{displaymath}
B_c = B_g.
\end{displaymath}](img1840.png) |
(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
![\begin{displaymath}
\oint {\bf H}\cdot d{\bf l} = \int{\bf j}_t\cdot d{\bf S} = N I.
\end{displaymath}](img1842.png) |
(883) |
Hence,
![\begin{displaymath}
(2\pi a-d) H_c + d H_g = N I.
\end{displaymath}](img1843.png) |
(884) |
It follows that
![\begin{displaymath}
B_g = \frac{N I}{(2\pi a-d)/\mu + d/\mu_0}.
\end{displaymath}](img1844.png) |
(885) |
Next: Magnetic energy
Up: Dielectric and magnetic media
Previous: Boundary conditions for and
Richard Fitzpatrick
2006-02-02