SHORTLY AFTER TAKEOFF THE PILOT INITIATED A LEFT CLIMBING TURN TO DEPART THE TRAFFIC PATTERN. DURING THE TURN THE AIRPLANE'S ENGINE STOPPED RUNNING. THE PILOT STATED HE ATTEMPTED TO RETURN TO THE AIRPORT BUT WAS NOT HIGH ENOUGH AND '... GOT TO SLOW...' TO LAND ON THE RUNWAY. THE AIRPLANE TOUCHED DOWN HARD IN A CORN FIELD AND NOSED OVER ACCORDING TO THE PILOT. AN ON-SCENE INVESTIGATION REVEALED WHAT THE FEDERAL AVIATION ADMINISTRATION'S PRINCIPAL MAINTENANCE INSPECTOR REFERRED TO AS 'TRACES' OF FUEL FOUND IN THE FUEL TANKS, CARBURETOR FLOAT BOWL, AND ACCELERATOR PUMP CHAMBER.