  Axion electrodynamics, first proposed in the context of particle physics,
manifests itself in condensed matter physics in the topological field theory
description of 3d topological insulators and gives rise to magnetoelectric
effect, where applying magnetic (electric) field $\mathbf{B}(\mathbf{E})$
induces polarization (magnetization) $\mathbf{p}(\mathbf{m})$. We use linear
response theory to study the associated topological current using the
Fu-Kane-Mele model of 3d topological insulators in the presence of
time-dependent uniform weak magnetic field. By computing the dynamical current
susceptibility $\chi^{\mathbf{j}_p\mathbf{j}_p}_{ij}(\omega)$, we discover from
its static limit an `order parameter' of the topological phase transition
between weak topological (or ordinary) insulator and strong topological
insulator, found to be continuous. The
$\chi^{\mathbf{j}_p\mathbf{j}_p}_{ij}(\omega)$ shows a sign-changing
singularity at a critical frequency with suppressed strength in the topological
insulating state. Our results can be verified in current noise experiment on 3d
TI candidate materials for the detection of such topological phase transition.
