He broke the fast at home before travelling, then he had intercourse with his wife; what does he have to do?

Firstly: 

The evidence from the Qur’an and Sunnah, and scholarly consensus, indicates that the traveller may break the fast of Ramadan, then make up the number of days that he did not fast. Allah, may He be exalted, says (interpretation of the meaning): “and whoever is ill or on a journey, the same number (of days which one did not observe Saum (fasts) must be made up) from other days” [al-Baqarah 2:185]. 

If a person is in his home, then he decides to travel, he is not called a traveller until he has passed beyond the built-up area of his city. So it is not permissible for him to avail himself of the concessions of travel, such as not fasting and shortening the prayers, just because he has the intention of travelling. That is because Allah, may He be exalted, has only made breaking the fast permissible for the traveller, and he is not a traveller until he has passed beyond (the boundary) of his city. 

Ibn Qudaamah said in al-Mughni (4/347), after mentioning that the one who travels during the day may break the fast: … it is not permissible for him to break the fast until he has left the houses behind, i.e., he has passed beyond them and emerged from the built-up area. Al-Hasan said: He may break the fast in his house, if he wishes, on the day that he is going to depart. Something similar was narrated from ‘Ata. Ibn ‘Abd al-Barr said: The view of al-Hasan is an odd (shaadhdh) view; no one who is not travelling has no right, on the basis of either rational thinking or reports, to break the fast; and the opposite was also narrated from al-Hasan. 

Then Ibn Qudaamah said: (That is also) because Allah, may He be exalted, says (interpretation of the meaning): “So whoever of you sights (the crescent on the first night of) the month (of Ramadan i.e. is present at his home), he must observe Sawm (fasts) that month” [al-Baqarah 2:185]. This one who is present at his home (and is not travelling) is not described as a traveller until he has left the city. So long as he is in the city, he comes under the heading of those who reside there (and are not travelling), hence he may not shorten the prayers. End quote. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked about a man who decided to travel, and he broke his fast at home because he was unaware (of the ruling), then he set out. Does he have to offer expiation? 

He replied: It is haraam for him to break the fast when he is in his house, but if he broke the fast before leaving his house, then he only has to make up the fast. End quote from Fataawa as-Siyaam, p. 133 

Secondly: 

The one who has intercourse during the day in Ramadan when he is fasting and not travelling has to offer a heavy expiation, which is to free a slave; if that is not possible then he has to fast for two consecutive months; if he is not able to do that, then he has to feed sixty poor persons. He also has to repent and make up that day. 

His wife also has to do the same if she was willing. It makes no difference whether ejaculation occurred or not; if intercourse, i.e., penetration occurred, then expiation must be offered. 

Thirdly: 

The kind of ignorance for which one may be excused is ignorance of the ruling. If a person failed to do an obligatory duty and did not know that it is obligatory, or he did something forbidden and did not know that it is forbidden, this is the one who is ignorant and may be excused for his ignorance. 

You may only be excused in your case if you did not know that this action of yours was haraam in the first place, or you thought that the one who has decided to travel can do that when he is still in his house. 

But in the case of one who knew that this action is forbidden, and he did it but was unaware of the punishment that results from it, this is not regarded as an excuse, because in this case the person went ahead and committed the sin and transgressed the sacred limits knowingly. 

Hence in the case of the Sahaabi who had intercourse with his wife during the day in Ramadan, the Prophet (blessings and peace of Allah be upon him) instructed him to offer expiation, and he did not let him off because of his ignorance of (the punishment), as was narrated by al-Bukhaari (1834) and Muslim (1111), because he had done it deliberately, knowing that it was forbidden. This was stated by al-Haafiz Ibn Hajar in al-Fath (4/207), based on the fact that he said “I am doomed” or “I am burnt”. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

If someone were to say: Wasn’t the man who came to the Messenger of Allah (blessings and peace of Allah be upon him) ignorant? 

The answer is: He was ignorant of what he was required to do (after breaking the fast); he was not ignorant of the fact that it was haraam. Hence he said “I am doomed.” If we say that ignorance is an excuse, we are not referring to ignorance of the consequences of this haraam action; rather what we are referring to is ignorance of whether this action is haraam or not. … Ignorance of the consequences of the haraam action is not an excuse; ignorance of whether the action is haraam or not is an excuse.

End quote from ash-Sharh al-Mumti‘, 6/417 

To sum up: 

You have to make up that day, and you have to offer expiation, so long as your breaking of the fast occurred when you had not yet begun your journey. 

Our advice to you is to try to fast the two consecutive months on cold or moderate days when the day is shorter and there will be less difficulty involved, or during the days of annual leave from work and other times when you have the opportunity to do what is required of you. If you are genuinely unable to fast, then it is permissible for you in that case to feed sixty poor persons, giving them one meal, or giving several meals until you complete the number required. 

Your wife also has to fast, and if she is not able to, she should feed sixty poor persons and not ten poor persons as mentioned in the question. 

See also the answer to question no. 106532

And Allah knows best.