The potential distribution in a plasma containing dust grains were derived where the Debye length can be larger or smaller than the average intergrain spacing. Three models were treated for the grain-plasma system, with the assumption that the system of dust and plasma is charge-neutral: a permeable grain model, an impermeable grain model, and a capacitor model that does not require the nearest neighbor approximation of the other two models. A gauge-invariant form of Poisson's equation was used which is linearized about the average potential in the system. The charging currents to a grain are functions of the difference between the grain potential and this average potential. Expressions were obtained for the equilibrium potential of the grain and for the gauge-invariant capacitance between the grain and the plasma. The charge on a grain is determined by the product of this capacitance and the grain-plasma potential difference.