why magnetic field due to infinitely long wire and finite long wire comes out to be same using amperes law?

Dear student,

If this is so, you are making a mistake while applying the Ampere’s law. Ampere’s law only states that line integral of magnetic field is equal to, which is always true.

Henceis always true.

However if you expanded it to, you have assumed yourself that magnitude of magnetic field is constant as well as ‘along’ the circular path, which is a wrong assumption. Hence by using you cannot proceed further from, and you would have to use biot-savart’s law.