We determine the effective dynamics for a spinless charged particle, in the presence of electromagnetic fields, constrained to a space curve. We employ the thin-layer procedure and a perturbative expansion for the Schr\"odinger equation is derived. We find that the first-order term in the perturbative expansion couples the dynamics of the normal and tangent degrees of freedom. However, there is always a gauge transformation that allows decouple the dynamics. We find that the effective Schr\"odinger equation in the curve contains an induced Zeeman coupling, independent of the curvature and torsion, which has not been previously reported. This coupling is characteristic of reducing the dimension by two and allows to identify an induced magnetic moment.