What is the ruling on a husband having intercourse with his wife a short time before the dawn (Fajr) adhan, where the couple discovered the entry of the time for Fajr after completing intercourse and did not hear the adhan, knowing that they completed their fast? And do they owe expiation?

If a husband and wife, believing that night was still present, engaged in intercourse, and then were surprised by the dawn breaking without having heard the Adhan, they are only required to make up the fast (qada') without incurring sin or expiation. This is because they did not intentionally violate the sanctity of the month, as Allah Almighty says: "And there is no blame upon you for that in which you have erred, but [there is blame for] what your hearts intended. And ever is Allah Forgiving and Merciful."