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

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.