  We study the deep inelastic scattering and photo-production modes of
$t\overline{t}$ pairs at the proposed LHeC and its potential to probe the
electromagnetic and weak dipole moments (MDM and EDM for $tt\gamma$) of the top
quark. A framework of eight independent gauge-invariant dimension-six operators
involving the top quark and the electroweak gauge bosons is used. Four of these
operators modify the charged $tbW$ coupling which can be probed through the
single (anti) top production mode as reported in the literature. One generates
$tt\gamma (Z)$ as well as $tbW$ couplings, while the other two do not generate
$tbW$ but only $tt\gamma (Z)$. Our focus is on the MDM and EDM of the top quark
for which the photo-production mode of $t\overline{t}$ can be an excellent
probe. At the proposed electron energies of $E_e =60$ and $140$ GeV the LHeC
could set constraints stronger than the indirect limits from $b\to s \gamma$
and the potential limits of the LHC through $t\overline{t} \gamma$ production.
