A mechanical analogy is used to analyze the interaction between the magnetic field, electric current, and deformation of interfaces in liquid metal batteries. In the framework of a low-mode, nondissipative, linear stability model, it is found that, during charging or discharging, a sufficiently large battery is prone to instabilities of two types. One is similar to the metal pad instability known to exist in the aluminum reduction cells. Another type is new. It is related to the destabilizing effect of the Lorentz force formed by the azimuthal magnetic field induced by the base current, and the current perturbations caused by the local variations of the thickness of the electrolyte layer.