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Dust polarization induced by aligned grains is widely used to study magneticfields in various environments, including star-forming regions. However, thequestion of to what optical depth grain alignment still exists in a densemolecular cloud (MC) is unclear. In this paper, we aim to achieve analyticalformulae for the minimum size of aligned grains ($a_{\rm align}$) androtational disruption ($a_{\rm disr}$) by RAdiative Torques (RATs) as afunction of the local physical parameters within dense MCs. We first find theanalytical approximations for the radiation strength and the mean wavelength ofthe attenuated radiation field in a dense MC without and with embedded starsand then derive analytical formulae for $a_{\rm align}$ and $a_{\rm disr}$ asfunctions of the visual extinction $A_{V}$ and the gas density. We find thatwithin a starless core of density $n_{\rm H}\sim 10^{5}\rm cm^{-3}$, grains ofsize $a<0.25\mu m$ can be aligned up to $A_{V}\sim 5$ by RATs, whereasmicron-sized grains can still be aligned at $A_{V}\sim 50$. The increase in thealignment size with $A_{V}$ can explain the presence of polarization holesobserved toward starless cores. For MCs with an embedded protostar, we findthat the efficiency of both alignment and rotational disruption increasestoward the protostar due to the increasing radiation strength. Such adisruption effect results in the decrease of the polarization degree with$A_{V}$ or emission intensity, which reproduces the popular polarization holesobserved toward the location of protostars. Finally, we derive the formula forthe maximum $A_{V}$ where grain alignment still exists in a starless core anddiscuss its potential for constraining grain growth.

Grain alignment and disruption by radiative torques in dense molecular clouds and implication for polarization holes