Magnetic contributions in Bekenstein type models

In this work, we analyze the spatial and time variation of the fine structure constant ($\alpha$) upon the theoretical framework developed by Bekenstein (Phys. Rev. D 66, 123514 (2002)). We have computed the field $\psi$ related to $\alpha$ at first order of the weak-field approximation and have also improved the estimation of the nuclear magnetic energy and, therefore, their contributions to the source term in the equation of motion of $\psi$. We obtained that the results are similar to the ones published in L. Kraiselburd and H. Vucetich, Int. J. Mod. Phys. E 20, 101 (2011) which were computed using the zero order of the approximation, showing that one can neglect the first order contribution to the variation of the fine structure constant. Through the comparison between our theoretical results and the observational data of the Eötvös-type experiments or the time variation of $\alpha$ over the cosmological time scale, we set constraints on the free parameter of the Bekenstein model, namely the Bekenstein length.