Hearing the Adhan being proclaimed after having sexual intercourse

Q: At dawn on Sunday (29/9/1416 A.H.) in the blessed month of Ramadan, I had sexual intercourse with my wife in a closed room at 5:17 am and we finished shortly at 5:22 am. The time of Fajr (Dawn) Prayer of Makkah was 5:31 am while the time of Fajr Prayer of Qunfudhah city was about 5:26 am on Sunday. When I finished and went out to take Ghusl (full ritual bath) I heard the Adhan (call to Prayer) of Fajr Prayer. Then, after about two minutes, the Mu'adhin (caller to Prayer) of the nearby mosque announced the Adhan. Am I sinful for having sexual intercourse with my wife at that time? It is noted that I did so another time in 1413 A.H. Really, I am (Part No. 9; Page No. 258) greatly perplexed, especially as my wife refused and she did so unwillingly. She suffers from a simple disease in her body. I hope you will give me a Fatwa (legal opinion issued a qualified Muslim scholar) for, as you know, this is a blessed month and we want to perfect our Sawm (Fast). It should be taken into consideration that my wife and I completed our Sawm and did not break it until that Sunday. May Allah reward you the best.

A: If the reality is what is mentioned, that after you finished sexual intercourse when you heard the Mu'adhin (caller to Prayer) and you became sure that some minutes remained between the occurrence of Fajr Prayer and sexual intercourse, then there is no blame on you, because sexual intercourse took place outside the time of Sawm. There is no blame on you for performing Ghusl after the beginning of the time of Fajr.May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.