We generalize a magnetogram-matching Biot-Savart law (BSL) from planar to spherical geometry. For a given coronal current density J, this law determines the magnetic field B ̃ whose radial component vanishes at the surface. The superposition of B ̃ with a potential field defined by a given surface radial field, Br, provides the entire configuration where Br remains unchanged by the currents. Using this approach, we (1) upgrade our regularized BSLs for constructing coronal magnetic flux ropes (MFRs) and (2) propose a new method for decomposing a measured photospheric magnetic field as B = B_pot + B_T + B_S ̃, where the potential, B_pot, toroidal, B_T , and poloidal, B_S ̃, fields are determined by Br, Jr, and the surface divergence of B − B_pot, respectively, all derived from magnetic data. Our B_T is identical to the one in the alternative Gaussian decomposition by Schuck et al. (2022), while B_pot and B_S ̃ are different from their poloidal fields B_P< and B_P>, which are potential in the infinitesimal proximity to the upper and lower side of the surface, respectively. In contrast, our B_S ̃ has no such constraints and, as B_pot and B_T, refers to the same upper side of the surface. In spite of these differences, for a continuous J–distribution across the surface, B_pot and B_S ̃ are linear combinations of B_P< and B_P>, which we derive in general form by representing these fields in terms of the corresponding Green’s functions. We demonstrate that, similar to the Gaussian method, our decomposition allows one to identify the footprints and projected surface-location of MFRs in the solar corona, as well as the direction and connectivity of their currents.