  The density dependence of the symmetry energy, instrumental in understanding
the behaviour of the asymmetric nuclear matter, is investigated within the
extended relativistic mean field (ERMF) model which includes the contributions
from the self and mixed interaction terms for the scalar-isoscalar ($\sigma$),
vector-isoscalar ($\omega$) and vector-isovector ($\rho$) mesons upto the
quartic order. Each of the 26 different parameterizations of the ERMF model
employed are compatible with the bulk properties of the finite nuclei. The
behaviour of the symmetry energy for several parameter sets are found to be
consistent with the empirical constraints on them as extracted from the
analyses of the isospin diffusion data. The neutron-skin thickness in the
$^{208}$Pb nucleus for these parameter sets of the ERMF model lie in the range
of $\sim 0.20 - 0.24$ fm which is in harmony with the ones predicted by the
Skyrme Hartree-Fock model. We also investigate the role of various mixed
interaction terms which are crucial for the density dependence of the symmetry
energy.
