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So, when the current carrying wire is infinitely long, we apply Ampere's circuital law. But if the wire has a certain length, then we use Biot-Savart law. My question is, what is wrong with using Ampere's law?

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Ampere's law is always true, but it is not always helpful. The expression for Ampere's law is:

$$ \oint\vec B\cdot d\vec l=\mu_0I$$

In the case of an infinite wire, we can make arguments, based on symmetries of the problem, that for a circular loop centered on the wire, $\vec B$ is constant and parallel to $d\vec l$, and so this expression simplifies to:

$$ 2\pi rB=\mu_0I\rightarrow B={\mu_0I\over2\pi r}$$

which gives us the magnitude of the $B$-field everywhere, without having to do any non-trivial integrals.

With a finite wire, there are fewer symmetries. You can still use Ampere's law, but without the symmetries it actually involves more math than just using the Biot-Savart law directly.

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