Exact solutions for internally induced magnetization in a shell

We analyse the external magnetic field generated by a spheroidal shell of c onstant susceptibility when it is magnetized by an internal magnetic field of arbitrary complexity. The analysis is relevant to the generation of the Earth's crustal magnetic field by the internal core field. We find an analy tical expression for such a crustal field that is expressed in an oblate sp heroidal coordinate system, valid for arbitrary flattening, shell thickness and susceptibility. Our exact calculation takes into account magnetization due to the magnetic field of the crust itself, generating an external fiel d that depends on terms both linear and non-linear in the susceptibility. E ach spheroidal harmonic coefficient of the external field is generated by a nd proportional to the same coefficient in the expansion of the inducing fi eld. The terms linear in the susceptibility are generated by the flattening of the Earth, and would otherwise vanish (by Runcorn's theorem) in the sph erical case. For a geophysically relevant model, the crustal field is weak, generated primarily by the non-linear terms, and dominated by long wavelen gths.