There is a use for curl - Faraday's Law!

Faraday's Law in integral form is the following: $$\oint_C \vec{E} \circ d\vec{l} = -\frac{d}{dt} \int_S \vec{B} \circ \hat{n}dA$$

It states that the circulation of an electric field $\vec{E}$ around a closed path $C$ is equal to the negative time derivative of the flux of the magnetic field through the surface $S$ bounded by $C$.

In other words, changing magnetic flux induce electric fields which form closed loops.