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As a new state of matter, the quantum liquid droplets formed in ultra-cold potassium (39K) atomic gases have been studied experimentally by Cabrera et al. (2018) and Semeghini et al. (2018). In this paper, we combined the experimental findings with the theory of mean-field, and obtained the potential function of inter-atomic interaction and the function of forces on atoms. The relation between the radial size of liquid droplet (σr) and magnetic field (B) is established based on the previous functions. From the viewpoint of energy balance, the condition under which the liquid droplets can stably exist and the dependence of the critical atom number (NC) on B are discussed. A semi-theoretical relation between NC and B is derived and verified to be valid by comparing the experimental values with the calculated ones. The energy balance (the potential energy of system equaling the kinetic energy of the system) can be considered as a condition under which the liquid droplets exist stably. Our results are helpful to further understanding of the quantum liquid droplets. Keywords: Quantum liquid droplets, Ultra-cold atomic gases, Bose-Einstein condensate, Interatomic potential