In this paper, we study the production of a top-quark in association with a bottom-quark and a electron-neutrino at the Future Circular Collider Hadron Electron (FCC-he) to probe the sensitivity on its magnetic moment ((a) over cap (V)) and its electromagnetic dipole moment ((a) over cap (A)) through the process e(-) p -> e(-) gamma p -> (t) over bart nu(e)bp. Assuming a large amount of collisions, as well as of data with cleaner environments, the FCC-he is an excellent option to study new physics, such as the (a) over cap (V) and (a) over cap (A). For our sensitivity study on (a) over cap (V) and (a) over cap (A), we consider center-of-mass energies root s = 7.07, 10 TeV and luminosities L = 50, 100, 300, 500, 1000 fb(-1). In addition, we apply systematic uncertainties delta(sys) = 0%, 3%, 5% and we consider unpolarized and polarized electron beam. Our results show that the FCC-he is a very good perspective to probe the (a) over cap (V) and (a) over cap (A) at high-energy and high-luminosity frontier. (C) 2019 The Author(s). Published by Elsevier B.V.