  In this paper we study the asymptotic behaviour of the quasilinear
$curl$-$curl$ equation of 3D magnetostatics with respect to a singular
perturbation of the differential operator and prove the existence of the
topological derivative using a Lagrangian approach. We follow the strategy
proposed in our recent previous work (arXiv:1907.13420) where a systematic and
concise way for the derivation of topological derivatives for quasi-linear
elliptic problems in $H^1$ is introduced. In order to prove the asymptotics for
the state equation we make use of an appropriate Helmholtz decomposition. The
evaluation of the topological derivative at any spatial point requires the
solution of a nonlinear transmission problem. We discuss an efficient way for
the numerical evaluation of the topological derivative in the whole design
domain using precomputation in an offline stage. This allows us to use the
topological derivative for the design optimization of an electrical machine.
