Electromagnetic Shielding - Mathematical Model

Mathematical Model

Suppose that we have a spherical shell of a (linear and isotropic) diamagnetic material with permeability, with inner radius and outer radius . We then put this object in a constant magnetic field:

Since there are no currents in this problem except for possible bound currents on the boundaries of the diamagnetic material, then we can define a magnetic scalar potential that satisfies Laplace's equation:

where

In this particular problem there is azimuthal symmetry so we can write down that the solution to Laplace's equation in spherical coordinates is:

After matching the boundary conditions

at the boundaries (where is a unit vector that is normal to the surface pointing from side 1 to side 2), then we find that the magnetic field inside the cavity in the spherical shell is:

where is an attenuation coefficient that depends on the thickness of the diamagnetic material and the magnetic permeability of the material:

This coefficient describes the effectiveness of this material in shielding the external magnetic field from the cavity that it surrounds. Notice that this coefficient appropriately goes to 1 (no shielding) in the limit that . In the limit that this coefficient goes to 0 (perfect shielding). In the limit that, then the attenuation coefficient takes on the simpler form:

which shows that the magnetic field decreases like .