  We study the lowest-order modifications of the static potential for
Born-Infeld electrodynamics and for the $\theta$-expanded version of the
noncommutative U(1) gauge theory, within the framework of the gauge-invariant
but path-dependent variables formalism. The calculation shows a long-range
correction ($1/r^5$-type) to the Coulomb potential in Born-Infeld
electrodynamics. However, the Coulomb nature of the potential (to order $e^2$)
is preserved in noncommutative electrodynamics.
