In general, the impact of geometric nonlinearity, which arises from geometric relationships governing the motion of constituent particles of elastic mediums, becomes critically important while the system operates under large deformations. In this case, the influence of different physics governing the system dynamics might be coupled with the impact of geometric nonlinearity. Here, for the first time, the non-zero component of the mean curvature tensor is nonlinearly expressed in terms of the middle-axis curvature of a cantilevered beam. To this aim, the concept of local displacement field together with inextensibility condition are employed. A nanowire-based capacitor is assumed to be excited by the electrostatic load that is composed of both DC and AC voltages. The main concern is on the case, in which it is necessary to polarize the electrodes with large amplitude voltages. Other physics, including surface strain energy, size-dependency, and dispersion force are modeled to predict the system response more accurately. Hamilton's principle is used to establish the motion equation, and the Galerkin method is applied to exploit a set of nonlinear ordinary differential equations (ODEs). Implementing a combination of shooting and arc-length continuation scheme, the frequency and force-displacement behaviors of the capacitor are captured near its primary resonance. The coupled effects of the nonlinear impact factor, surface elasticity and size parameters on the bifurcation point's loci and dynamic pull-in instability are studied.