The trick is that the area of my hand doesn't matter. The flux has to pass through a metal loop so we are only interested in the flux passing through the ring itself. I used http://www.ngdc.noaa.gov/geomag-web/#igrfwmm to calculate the strength of the Earth's magnetic field for 55 degrees north and 3 degrees west. It gave me 49,623 nanoTeslas. The area through the middle of the ring through which flux can pass: well, radius is 1cm so area = 3.1 x 10^-4 square metres. Flux = field strength x area = 1.6 x 10^-8Wb. Flux linkage = flux x number of turns. There is only one turn here, so flux linkage = 1.6 x 10^-8 Wb(turns). By Faraday's Law, emf = rate of change of flux linkage. If I move my hand through 90 degrees, I go from full flux to no flux through the ring. It takes 0.1s. Hence emf = 1.6 x 10^-7V.
This emf makes a current flow round my ring. The resistance of my ring = (resistivity x circumference) / cross sectional area of the metal itself. Gold has a resistivity of 2.44 x 10^-8 Ohmmetres. The ring is 5mm deep by 1mm wide so cross-sectional area = 5 x 10^-6 square metres. I calculate a resistance of 3 x 10^-4 Ohms.
Current = emf/resistance so a current of 5 x 10^-4A flows. Not much to worry about there.
The only thing to add is that I have assumed you knew I was standing facing north, in line with the field.
