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# Magnetic Shielding

There are many situations, particularly in experimental physics, where it is desirable to shield a certain region from magnetic fields. This goal 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 mu-metal, and is an alloy of 5 percent copper, 2 percent chromium, 77 percent nickel, and 16 percent iron. The maximum permeability of mu-metal is about . This material also possesses a particularly low retentivity and coercivity. Unfortunately, mu-metal 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 . Because there are no free currents in the problem, we can write . Furthermore, because 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

 (743) (744)

Let us try the following test solution for the magnetic potential:

 (745)

for ,

 (746)

for , and

 (747)

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. Because there is a uniqueness theorem associated with Poisson's equation (see Section 2.3), we can be certain that this potential is the correct solution to the problem provided that the arbitrary constants , , et cetera, can be adjusted in such a manner that the boundary conditions at and are also satisfied.

The continuity of at and requires that

 (748)

and

 (749)

The boundary conditions (744) and (745) yield

 (750)

and

 (751)

It follows that

 (752) (753) (754) (755)

Consider the limit of a thin, high permeability shell for which , , and . In this limit, the field inside the shell is given by

 (756)

Thus, given that for mu-metal, 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. Note, however, that as the external field-strength, , is increased, the mu-metal shell eventually saturates, and gradually falls to unity. Thus, extremely strong magnetic fields (typically, tesla) are hardly shielded at all by mu-metal, or similar magnetic materials.

Next: Magnetic Energy Up: Magnetostatics in Magnetic Media Previous: Soft Iron Sphere in
Richard Fitzpatrick 2014-06-27