  A study of the top quark \emph{Flavour Changing Neutral Current} (FCNC)
through $Z$-boson has been performed in the proposed future $e^-p$ collider for
the energy, $E_{e(p)} = 60~(7000)$~GeV. We considered an effective theory where
the anomalous FCNC couplings are of vector and tensor nature. The effect of
these couplings is probed in the single top production along with the scattered
electron. The polar angle $\theta$ of the electrons coming out of the primary
vertex in association with the top quark polarization asymmetries constructed
from the angular distribution of the secondary lepton arising from the top
decay, allow to distinguish the Lorentz structure of the coupling. From a
multi-parameter analysis, we obtain a reach of ${\cal O} (10^{-2})$ in the case
of $Ztu$ and $Ztc$ couplings at an integrated luminosity of 2~ab$^{-1}$ at 95\%
C.L.
