In this paper, we numerically investigate the near-field optic properties and surface plasmon resonance effects on a pair of finite height gold nanorod and a pair of finite height core-shell gold nanorod by means of finite element method with three-dimensional calculations. Numerical results of resonant wavelengths corresponding to the effects of different illumination wavelengths, thickness of core-shell, electric field components, electrical field stream lines, charge distributions, energy flow, and the core relative permittivities of a pair of finite height core-shell gold nanorod are also reported. These results are crucial in designing localized surface plasmon resonance (LSPR) sensors and other optical devices based on a pair of finite height core-shell gold nanorod interacting with an incident light.