Response of a permeable ellipsoid to an imposed magnetic gradient – implications for borehole measurements of the magnetic gradient tensor inside magnetic bodies

Summary

This paper discusses the internal field and gradient tensor components for a permeable ellipsoidal body in a magnetic medium with an applied non-uniform magnetic field. The formulae can also be applied to the field and gradients measured in an ellipsoidal cavity within a magnetic medium. A long, narrow cylindrical cavity, representing a borehole that penetrates a magnetic body, is a particularly useful special case. I give explicit expressions for the internal gradient tensor within a permeable triaxial ellipsoid placed in a uniform gradient. Expressions are given for the shielding of applied gradients of all orders for the special case of a spherical body and for the amplification of applied gradients of all orders in a spherical cavity within a magnetic medium.

Magnetic sensors cannot measure the internal field and gradient components of a magnetic body directly, as the measuring instrument must be placed within a cavity, such as a borehole, within the magnetic material. This modifies the measured field and gradients. I give expressions for correcting measured gradient tensor components in a borehole within a magnetic body.

• borehole magnetics
• magnetic gradient tensor
• ellipsoid