Electric-field noise due to surfaces disturbs the motion of nearby trapped ions, compromising the fidelity of gate operations that are the basis for quantum computing algorithms. We present a method that predicts the effect of dielectric materials on the ion's motion. Such dielectrics are integral components of ion traps. We find quantitative agreement between a model with no free parameters and measurements of a trapped ion in proximity to dielectric mirrors. We expect that this approach can be used to optimize the design of ion-trap-based quantum computers and network nodes.