  In this article, we develop and analyze a finite element method with the
first family N\'ed\'elec elements of the lowest degree for solving a Maxwell
interface problem modeled by a $\mathbf{H}(\text{curl})$-elliptic equation on
unfitted meshes. To capture the jump conditions optimally, we construct and use
$\mathbf{H}(\text{curl})$ immersed finite element (IFE) functions on interface
elements while keep using the standard N\'ed\'elec functions on all the
non-interface elements. We establish a few important properties for the IFE
functions including the unisolvence according to the edge degrees of freedom,
the exact sequence relating to the $H^1$ IFE functions and the optimal
approximation capabilities. In order to achieve the optimal convergence rates,
we employ a Petrov-Galerkin method in which the IFE functions are only used as
the trial functions and the standard N\'ed\'elec functions are used as the test
functions which can eliminate the non-conformity errors. We analyze the inf-sup
conditions under certain conditions and show the optimal convergence rates
which are also validated by numerical experiments.
