In spintronic terahertz (THz) emitters, THz radiation is generated by exciting an ultrafast spin current through femtosecond laser excitation of a ferromagnetic-nonmagnetic metallic heterostructure. Although an extensive phenomenological knowledge has been built up during the last decade, a solid theoretical modeling that connects the generated THz signal to the laser induced-spin current is still incomplete. Here, starting from general solutions to Maxwell's equations, we model the electric field generated by a superdiffusive spin current in spintronic emitters, taking Co/Pt as a typical example. We explicitly include the detector shape, which is shown to significantly influence the detected THz radiation. Additionally, the electron energy dependence of the spin Hall effect is taken into account, as well as the duration of the exciting laser pulse and the thickness of the detector crystal. Our modeling leads to realistic emission profiles and highlights the role of the detection method for distinguishing key features of the spintronic THz emission.