current in Born-Infeld electrodynamics

I have a question about currents in Born-Infeld electrodynamics. To fix the notation, the Born-Infeld Lagrangian is $$\mathcal{L} = - \sqrt{- \det \left(\eta_{\mu\nu} + F_{\mu\nu}\right)}$$ I calculate the electric field of a current $I$ flowing in the positive direction on the $z$-axis. Solving the equations of motion coming from the Lagrangian above, I get $$\vec H = \frac{I}{2 \pi s} \vec{e}_{\phi}$$ and $$\vec{B} = \dfrac{\vec H}{\sqrt{1 - H^2}}$$ I have used the usual cilindrical coordinates: $s$ is the distance to the $z$ axis and $\phi$ is the angle. From this it is clear that $H$ should satisfy $H^2 < 1$ and thus $\dfrac{I}{2 \pi s} < 1$.

This problem is also solved by Gibbons , who remarks that Presumably an electric current $J_e$ cannot, according to Born- Infeld theory, be contained within a wire of radius less than $2 J_e$.

I am puzzled by this result. Does anybody have insight in what is going on? Suppose the world was described by Born-Infeld instead of Maxwell, what would happen if one increases the current inside a wire? What would make it impossible to increase the current above a certain value?

 Gibbons, Born-Infeld particles and Dirichlet p-branes, page 13 https://arxiv.org/abs/hep-th/9709027

• Ok, I agree with your comment. I have removed the word "physically" from my question and added the clarification "suppose the world was described by Born-Infeld instead of Maxwell." – wiskundeliefhebber Feb 12 '17 at 18:06