Magnetic Shielding Effect of a Spherical Shell
Consider a spherical shell of linear magnetic material with relative permeability placed in a uniform magnetic field . The magnetic fields in this region can be described by a magnetic potential . Selecting the direction of as the axis of spherical coordinates , is given by , where is a Legendre function. The magnetic field at any point is . The coefficients , in the regions with (1) , (2) , and (3) are determined by considering the boundary conditions at and , taking into account the permeability in each region: (1) and (3) and (2) . As the result, the field in (1) is that of superimposed with the contribution of a magnetic dipole. The field in (3) turns out to be uniform, with magnitude considerably lower than . The magnetic induction or B field is obtained by , where or depending on the region.
The direction of is chosen as the axis. The fields are displayed on the cross section through the center of the spherical shell. The color represents the field intensity normalized by . The field lines are calculated from equally distributed points. You can vary the shell's relative permeability and its configuration parameters , . It is observed that the field intensity in the inner space is greatly reduced relative to that in the outer space, especially for a large . This is shown in snapshots 1 and 2. Since the E and B fields are shown in normalized bases, the two fields are equal in free space, but the E field is much smaller in the shell (magnetic medium), as shown in snapshot 3.