This work proposes a micromechanical model to address the electrical contact problem between a spherical indenter and a homogeneous half-space, considering the coupling competition between frictional heat and Joule heating. Unlike methods that determine undetermined coefficients in the general solution of partial differential equations, this research leverages Green"s functions and Eshelby tensor theory to reveal the transfer function relationships among current, heat flux, and stress, providing an efficient approach for solving sequential electro-thermal-mechanical multiphysics coupling problems. Through integral transformations of complex potential functions, explicit frequency-domain solutions for thermal and elastic fields are derived. The model is validated by finite element simulations, systematically investigating the effects of speed, load, voltage, current, and indenter radius on the coupling competition between frictional and Joule heating. This work offers new thermal management insights for minimizing the maximum temperature rise.