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Although charging of insulators by atomic force microscopy (AFM) has found widespread interest, often with data storage or nanoxerography in mind, less attention has been paid to the charging mechanism and the nature of the charge. Here we present a systematic study on charging of amorphous polymer films by voltage pulses applied to conducting AFM probes. We find a quadratic space charge limited current law of Kelvin probe force microscopy and electrostatic force microscopy peak volumes in pulse height, offset by a threshold voltage, and a power law in pulse width of positive exponents smaller than one. We interpret the results by a charging mechanism of injection and surface near accumulation of aqueous ions stemming from field induced water adsorption, with threshold voltages linked to the water affinities of the polymers.