It has nothing to do with current, it is a reaction from air and moisture called patina.
Initially, bare Cu metal atoms react with air to form the pink oxide, cuprite, Cu2O, which has Cu+1 cations. This gradually oxidizes further to the black oxide, tenorite, CuO, with Cu+2 ions. The black sulfide CuS also sometimes forms. In the presence of moisture, the blackish layer slowly reacts with sulfur dioxide and carbon dioxide from the air to eventually form the patina, which is a mixture of 3 minerals:
brochantite, a green, hydrated copper sulfate, Cu4SO4(OH)6
malachite, the green, hydrated copper carbonate Cu2CO3(OH)3
azurite, the blue, hydrated copper carbonate Cu3(CO3)2(OH)2
In these compounds, copper atoms from the metal surface have been fully oxidized into Cu+2 (cupric) ions. The rate of patina formation and the proportion of the components depend upon humidity and the amount of air pollution.