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The process of a dust-grain charging in developing air plasma with an external source S (cm-3 s-1) is studied analytically and numerically in drift-diffusion approximation. It is shown that, when the source is turned on, the steady-state level of electron concentration in the bulk plasma is attained rather quickly (the time being determined by the attachment rate). The time of establishing the ion concentration depends on the source intensity as S-12/. In the steady state, the ratio of ion-to-electron concentrations is proportional to the source intensity as S-12/ and can be very large for low-intensity sources. As a consequence, the process of a dust-grain charging in developing plasma differs significantly for low- and high-intensity sources. At sufficiently high rates of ionization, the grain charge in air is comparable to that in electropositive gases. For low-intensity sources when the ion concentration exceeds that of electron by several orders of magnitude, the grain charge is of the order of several tens of electron charges. The oscillating regime of a dust-grain charging is observed for low-intensity sources.