A two‐dimensional time‐dependent model of polar cap arcs has been developed. The electrodynamics of the polar cap arcs are treated self‐consistently in the frame of the coupled magnetosphere‐ionosphere system. The focus of this paper is to introduce the physics and mathematical formulation of the model and describe the features of the spatial structure and temporal evolution of the polar cap arcs. The modeling results indicate that the time constant for the formation of the polar cap arcs is around 10 min. It is found that an initial single‐arc precipitation pattern associated with a polar cap arc tends to split into multiple precipitation regions and leads to a multiple structure of the polar cap arcs. It is also found that strong downward field‐aligned currents can develop near the intensive upward field‐aligned currents and form a pair structure of the field‐aligned current in the polar cap arcs. The model predicts the existence of plasma flow crossing the polar cap arcs, but the amplitude of such a flow is small, and the characteristic time scale of it is much larger than the time constant for the formation of the polar cap arcs. Our results also show that while polar cap arcs are developing, the associated current system tends to become further localized and at steady state the current system in the narrow regions around the arcs is locally self‐closed.