Recent magnetic and optical observations of the cusp ionosphere indicate the presence of elongated plasma clouds (EPCs) in the magnetosphere. The formation of EPCs may originate from either flux transfer events or plasmas injected into the magnetosphere. Based on observations, a self-consistent current system in the cusp ionosphere is modeled for elongated plasma clouds, which are allowed to move in any given direction. The current system consists of the Pedersen current, the Hall current, and the field-aligned current. The field-aligned current flows upward on one-half of an elongated plasma cloud and downward on the other half. It is found that the distribution of the field-aligned current of EPCs is quite different from the case with a circular cross-section, which has been studied previously. The plasma convection pattern and the convection electric field are also obtained. The current and convection patterns are highly dependent on the orientation of the convection velocity. The magnetic field on the ground produced by this current system is calculated. The total field-aligned current and profiles of the ground magnetic field are obtained. By assuming that the current system moves with a velocity V0~1--4 km/s in the cusp ionosphere, the variation of the perturbed ground magnetic field as a function of time is obtained. It is found that, for a moving plasma cloud with a circular cross-section, only unipolar profiles of the vertical magnetic field can be observed by a ground observer. However, for a moving plasma cloud with an elongated cross-section, both unipolar and bipolar profiles can be observed.