بسم الله الرحمن الرحيم
(Fatwa: 880/880/M=1434)
It is unlawful and sin to swear in the name of other than Allah. You should not have sworn in the name of your children in any case. Tell this ruling to your wife and avoid it in future. In this case there is no financial kaffarah on you. Do taubah and istighfar.
Allah knows Best!
Darul Ifta,
Darul Uloom Deoband
