The Freedericksz transition threshold is investigated in a homogeneously aligned carbon nanotube
(CNT) doped liquid crystal (LC) planar cell. By neglecting the aggregate and bundle formation of
CNTs in a low concentration of CNT doped LC cell, the reduction in the Freedericksz transition
threshold is calculated theoretically. Flexoelectric polarisation tensorial expansion in terms of the
order parameter and the generalised Helfrich theory is used to estimate the flexoelectric coefficients
in the mixture. Dipolar and quadrupolar polarisations are studied in the flexoelectric
coefficients of the mixture. It is shown that doping CNTs changes the flexoelectric coefficients
through the order parameter of LCs. According to these calculations, for the LC molecules with
a dipole moment parallel to their molecular axis, up to 4-fold increase in flexoelectric coefficient is
expected, which is in good accordance with experimental reports. Also, it is shown that in the
absence of dipolar flexoelectricity, the quadrupolar polarisation is responsible for the changes of
the flexoelectric coefficients.