  We extend the usual world-volume action for a Dp-brane to the case of N
coincident Dp-branes where the world-volume theory involves a U(N) gauge
theory. The guiding principle in our construction is that the action should be
consistent with the familiar rules of T-duality. The resulting action involves
a variety of potential terms, i.e., nonderivative interactions, for the
nonabelian scalar fields. This action also shows that Dp-branes naturally
couple to RR potentials of all form degrees, including both larger and smaller
than p+1. We consider the dynamics resulting from this action for Dp-branes
moving in nontrivial background fields, and illustrate how the Dp-branes are
``polarized'' by external fields. In a simple example, we show that a system of
D0-branes in an external RR four-form field expands into a noncommutative
two-sphere, which is interpreted as the formation of a spherical D2-D0 bound
state.
