  We present a novel \textit{a posteriori} error estimator for N\'ed\'elec
elements for magnetostatic problems that is constant-free, i.e. it provides an
upper bound on the error that does not involve a generic constant. The
estimator is based on equilibration of the magnetic field and only involves
small local problems that can be solved in parallel. Such an error estimator is
already available for the lowest-degree N\'ed\'elec element [D. Braess, J.
Sch\"oberl, \textit{Equilibrated residual error estimator for edge elements},
Math. Comp. 77 (2008)] and requires solving local problems on vertex patches.
The novelty of our estimator is that it can be applied to N\'ed\'elec elements
of arbitrary degree. Furthermore, our estimator does not require solving
problems on vertex patches, but instead requires solving problems on only
single elements, single faces, and very small sets of nodes. We prove
reliability and efficiency of the estimator and present several numerical
examples that confirm this.
