In this article, we present an analytical model to calculate the perturbation of the magnetic vector potential due to a non-conducting inclusion in a liquid metal translated at constant velocity. The model is valid for a low magnetic Reynolds number, Rem, and a low shielding parameter, Sω. We establish that the first order perturbation of the vector potential can be directly obtained from the value of the unperturbed eddy-current calculated at the location of the inclusion. An ECFM (Eddy-Current FlowMeter) device has been designed to validate the model and to study the frequency effects of the eddy-current and the volume of one inclusion and its location. We observe a good agreement between the model predictions and experimental data within the limit of the validity of the model. Based on scaling analysis, we show that a unique relation exists between the perturbation of the emf (electromotive force) measured with the ECFM and the volume of the inclusion and its location, which reflects the self-similarity of the model solution.