Closed-form expressions of vector gravity and magnetic field due to a rectangular disk.

In the case of applying magnetic exploration to detect underground man-made objects precisely, it is important to calculate magnetic responses analytically due to various shapes, such as one-dimensional line segments, 2D disk types, and 3D prismatic bodies. As part of these contributions, in this study, I derive the closed-form expressions of the magnetic field of one of 2D disk types, a rectangular disk. First, the gravitational potential due to a rectangular disk parallel to the x-y plane is defined by the two-dimensional surface integral. The vector gravity can be derived by differentiating the gravitational potential in each axial direction. The surface integrals that include the multiple square roots of the distance between observation points to the rectangular disk are required. Differentiating the vector gravity once more in each axial direction yields the gravity gradient tensor. For a causal body with constant magnetization, Poisson's relation is applied to convert the gravity gradient tensor to the magnetic field. The derived expressions of magnetic response are validated by comparing them with a three-dimensional rectangular prism with thin thickness. For the inclined rectangular disk, the magnetic fields are computed by transforming the observing coordinate system to the coordinate system affixed to the rectangular disk, and then the magnetic fields can be obtained by the inverse coordinate transformation.