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There are many situations, particularly in experimental physics, where
it is desirable to shield a certain region from magnetic fields. This
can be achieved by surrounding the region in question by a material of
high permeability. It is vitally important that a material used as a magnetic
shield does not develop a permanent magnetization in the presence
of external fields, otherwise the material itself
may become a source of magnetic fields. The most effective
commercially available magnetic shielding material is called
Mumetal, and is an alloy of 5% Copper, 2% Chromium, 77% Nickel,
and 16% Iron. The maximum permeability of Mumetal is about .
This material also possesses a particularly low retentivity and coercivity.
Unfortunately, Mumetal is extremely expensive. Let us investigate how
much of this material is actually required to shield a given region
from an external magnetic field.
Consider a spherical shell of magnetic shielding, made up of material of
permeability , placed in a formerly uniform magnetic field
. Suppose that the inner radius of the
shell is and the outer radius is . Since there are no free
currents in the problem, we can write
.
Furthermore, since
and
, it is clear that the magnetic scalar potential satisfies Laplace's
equation,
, throughout all space. The boundary conditions
are that the potential must be well behaved at and
,
and also that the tangential and the normal components of and
, respectively, must be continuous at and .
The boundary conditions on merely imply that the scalar potential
must be continuous at and . The boundary
conditions on yield
Let us try the following test solution for the magnetic potential:

(600) 
for ,

(601) 
for , and

(602) 
for . This potential is certainly a solution of Laplace's equation
throughout space. It yields the uniform
magnetic field as
, and satisfies physical boundary conditions at and infinity.
Since there is a uniqueness theorem associated with Poisson's equation,
we can be certain that this potential is the correct solution to the
problem provided that the arbitrary constants , ,
etc. can be adjusted in such a manner that the boundary
conditions at and are also satisfied.
The continuity of at and requires that

(603) 
and

(604) 
The boundary conditions (3.175) yield

(605) 
and

(606) 
It follows that
Consider the limit of a thin, high permeability shell for which
, , and
. In this limit, the
field inside the shell is given by

(611) 
Thus, if
for Mumetal, then we can reduce the magnetic
field strength inside the shell by almost a factor of 1000 using a shell
whose thickness is only 1/100th of its radius. Clearly, a little Mumetal
goes a long way! Note, however, that as the external field strength,
, is increased, the Mumetal shell eventually saturates, and
gradually falls to unity. Thus, extremely strong magnetic
fields (typically,
tesla) are hardly shielded
at all by Mumetal, or similar magnetic materials.
Next: Magnetic energy
Up: The effect of dielectric
Previous: A soft iron sphere
Richard Fitzpatrick
20020518