Part 1: The Earth's magnetic field strength in the orbital coordinate system. A charged satellite moves on a Keplerian orbit in the Earth's magnetic field (EMF). With the aim to construct the sufficiently precise mathematical model of attitude motion of a satellite, the EMF intensity H is determined on the basis of refined, namely a quadrupole model of geomagnetic potential in view of the EMF nonuniformity in the volume of the satellite. The general problem of determining the gradient of a vector field by the known scalar potential prescribed in spherical coordinates is solved. The components of vector H in the orbital coordinate system are found. It is shown that accounting the quadrupole component of geomagnetic potential contributes greatly to determining EMF intensity.