# Under what conditions can we use Ampere’s law to find magnetic ﬁelds?

I'm trying to understand the different formulas to calculate magnetic ﬁelds, it looks like Ampere’s law is an easier way than the Biot-Savart Law, but only works under certain conditions (similar to Gauss’ law and Coulomb’s law in electrostatics). Under what conditions can we use Ampere’s law instead of the Biot-Savart law?

Well... under all conditions, just like Gauss' law. All of those laws are correct for not-too-weird configurations, but sometimes using a certain law can be more convenient. So, assuming you meant something more like, when is Ampère's law is more convinient to use than Biot-Savart, the answer is: When you can locate a path with constant magnetic field amplitude and you can measure its length.

The easiest examples are if you know $$\mathbf B$$ is

1. Cylindrically symmetric (e.g. infinite straight wires with radial current profile: $$\mathbf j(\mathbf r) = \mathbf j(\|\mathbf \rho\|)$$) where you pick a circle as your path or

2. if it is constant along a sheet (e.g. infinite wire with infinite width) where you select a path that goes along $$\mathbf B$$ for a while, goes through the current sheet and again does the same.

In such cases, you can find $$\mathbf B$$ because $$\hat{\mathbf{B}} = \hat{\mathbf{\ell}}$$ and thus from

$$\int_\mathcal C \mathbf{B}\cdot\hat{\mathbf{\ell}} = L\,\|B\| = \mu_0I_{enc}$$

you know the amplitude of $$\mathbf{B}$$. You know the direction is $$\hat{\mathbf{\ell}}$$ anyway so q.e.d.

In magnetostatics, the two laws are equivalent, so you can in principle use both of them. However, applying the integral form of Ampere's law to calculate magnetic fields is straightforward only in the simplest of geometries: ones with symmetries to allow easy evaluation of the integral. The most famous example is the magnetic field due to an infinite straight wire, in which case you can exploit the symmetry along and about the wire.

In addition, with Ampere's law, you may need to consider the entire circuit into account when calculating magnetic fields: trying to apply it to a finite wire can lead to contradictory results (see this answer). This is because currents at steady state always flow in a loop, and can't solely exist in a finite wire that doesn't form a loop. The Biot-Savart law gives you a way to speak of the magnetic field contribution due to a piece of the circuit, although you would still need to integrate along the entire circuit to calculate the total magnetic field at a given point.