Born and Infeld introduced in 1934 a non linear version of the Maxwell equations, which is still for use in high energy physics. Remarkably enough, the Born-Infeld system can be enlarged as a 10x10 system of hyperbolic conservation laws, quite similar to the classical MHD equations, with a nearly quadratic conserved energy. This allows us to perform some asymptotic analysis by using a relative entropy method going back to Dafermos.