Metal additive manufacturing (MAM) has gained extensive attention from research institutes and companies to fabricate intricate parts and functionally graded materials. With the development of MAM, scientists also try to implement magnetic fields into Laser Powder Bed Fusion (L-PBF), to control the molten pool dynamics, tailor the dendritic morphology and improve the mechanical property of the as-built part. To study the molten pool dynamics under an external magnetic field, a Thermoelectric Magnetohydrodynamic (TEMDH) model is developed by incorporating the electrodynamic model with the Seebeck effect into the multi-physics thermal-fluid flow model. The Seebeck effect in the molten pool is analyzed with the simulation of stationary laser melting on a bare plate. It shows that the Lorentz force is relatively high at the bottom of the molten pool, and the magnitude is comparable to the buoyancy force density, which can influence the molten pool dynamics, especially in deep keyhole mode without external magnetic fields. Under the external magnetic fields, the Lorentz force densities under the external magnetic fields reach 1.5e8 N/m^3 and are about four orders of magnitude higher than that without an external magnetic. Such Lorentz force density is sufficient to make a difference in the molten pool shape. Applying the TEMHD models to the L-PBF, the laser scanning simulations show that although the Lorentz forces under the y- and z-direction external magnetic fields distribute differently, both of them can suppress the fluid circulation and decrease the fluid spiral rotating behind the keyhole in the molten pool. Moreover, by validating against experimental results, the Lorentz force at the bottom of the molten pool is strong enough to transit the columnar grains to equiaxed grains under external magnetic fields. Thus, an external magnetic field can be a useful tool to control the molten pool flow and microstructure evolution in the L-PBF process.