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This paper presents an alternative method to model a multilayer voice coil or an air-cored electromagnet as an equivalent permanent magnet (ePM) such that its magnetic field can be characterized by a distributed set of multipoles model. We validate the ePM model by comparing the computed results against exact solutions and illustrate its effectiveness in computing the magnetic force using the dipole force equation. Unlike methods that are based on the Lorentz force equation or the Maxwell stress tensor, which require computing the volume or surface integrals to derive the forces, the closed-form dipole force method that replaces integrations with summations dramatically reduces computation time. We compare the dipole force computation against results of the Lorentz force equation and the Maxwell stress tensor method, and validate the comparisons against published experimental data. To demonstrate the effectiveness of the method, we compute the inverse torque model of a 3-DOF orientation stage operated on the principle of a spherical motor that has more controlling inputs than its mechanical DOF.