Frame dragging (Lense-Thirring effect) is generally associated with rotating astrophysical objects. However, it can also be generated by electromagnetic fields if electric and magnetic fields are simultaneously present. In most models of astrophysical objects, macroscopic charge neutrality is assumed and the entire electromagnetic field is characterized in terms of a magnetic dipole component. Hence, the purely electromagnetic contribution to the frame dragging vanishes. However, strange stars may possess independent electric dipole and neutron stars independent electric quadrupole moments that may lead to the presence of purely electromagnetic contributions to the frame dragging. Moreover, recent observations have shown that in stars with strong electromagnetic fields, the magnetic quadrupole may have a significant contribution to the dynamics of stellar processes. As an attempt to characterize and quantify the effect of electromagnetic frame dragging in these kinds of astrophysical objects, an analytic solution to the Einstein-Maxwell equations is constructed here on the basis that the electromagnetic field is generated by the combination of arbitrary magnetic and electric dipoles plus arbitrary magnetic and electric quadrupole moments. The effect of each multipole contribution on the vorticity scalar and the Poynting vector is described in detail. Corrections on important quantities such the innermost stable circular orbit (ISCO) and the epicyclic frequencies are also considered.