In this paper we address the question of the influence of cathode tip geometry on the physical quantities relevant to Tungsten Inert Gas (TIG) arc welding. Although cathode geometry is known to influence arc-plasmas, a thorough quantification of the Lorentz phenomenon, which is the dominant source of momentum in the fluid flow of TIG arcs has not been strongly investigated. Therefore, we have performed a numerical parametric study at a constant inlet electric current and arc height, for different cathode sizes and shapes (pointed, chamfered and rounded tips), parameterised by the truncation angle and tip radius. To this end, we developed a coupled steady-state magneto-hydrodynamic finite element model in a 2D axi-symmetric configuration, implemented in the Cast3M toolbox. The paper characterises, quantifies and analyses the influence of tip geometry on TIG arcs and the workpiece. The model reveals that tip size and truncation angle similarly influence the magnitude of the transported quantities of the arcs. We find that the imparted momentum and transferred heat flux to the workpiece vary about 4 and 2-fold between the bluntest and sharpest cathodes, respectively. This signifies the importance of the choice of cathode tips in numerical TIG welding simulations. Agreement between the literature and the current work is analysed and discussed and the importance of coupling the cathode and arc domains is highlighted.