The Born-Infeld system is a nonlinear version of Maxwell’s equations. We first show that, by using the energy density and the Poynting vector as additional independent variables, the BI system can be augmented as a 10x10 system of hyperbolic conservation laws. The resulting augmented system has some similarity with MHD equations and enjoy remarkable properties (existence of a convex entropy, galilean invariance, full linear degeneracy). In addition, the propagation speeds and the characteristic fields can be computed in a very easy way, in contrast with the original BI equations. Then, we investigate several limit regimes of the augmented BI equations, by using a relative entropy method going back to Dafermos, and recover, the Maxwell equations for low fields, some pressureless MHD equations (describing string motion) for high fields, and pressureless gas equations for very high fields.