d.c. Look up d.c. at Dictionary.com
abbreviation of direct current, attested from 1898.
D.C. Look up D.C. at Dictionary.com
abbreviation of District of Columbia, apparently not widely used before 1820, but eventually it became necessary to distinguish it from the many other Washingtons in America. The city and the district (then known as Territory of Columbia) were named 1791; the territory was organized as a "district" in 1801, but the towns within it (Washington, Georgetown, Alexandria) remained separate municipalities and at one time all took D.C. The district was effectively organized as a unitary municipality in 1871.