This paper presents an analytical model for the force and torque developed by a reaction sphere actuator for satellite attitude control. The reaction sphere is an innovative momentum exchange device consisting of a magnetic bearings spherical rotor that can be electronically accelerated in any direction making all the three axes of stabilized spacecrafts controllable by a unique device. The spherical actuator is composed of an 8-pole permanent magnet spherical rotor and of a 20-coil stator. Force and torque analytical models are derived by solving the Laplace equation and applying the Lorentz force law. The novelty consists in exploiting powerful properties of spherical harmonic functions under rotation to derive closed-form linear expressions of forces and torques for all possible orientations of the rotor. Specifically, the orientation of the rotor is parametrized using seven decomposition coefficients that can be determined noniteratively and in a linear fashion by measuring the radial component of the magnetic flux density from at least seven different locations. Therefore, force and torque models for all possible orientations of the rotor are expressed in closed form as linear combination of mutually orthogonal force and torque characteristic matrices, which are computed offline. The proposed analytical models are experimentally validated using a developed laboratory prototype.