Cold ions trapped in the vicinity of conductive surfaces experience heating of their oscillatory motion. Typically, the rate of this heating is orders of magnitude larger than expected from electric field fluctuations due to thermal motion of electrons in the conductors. This effect, known as anomalous heating, is not fully understood. One of the open questions is the heating rate's dependence on the ion-electrode separation. We present a direct measurement of this dependence in an ion trap of simple planar geometry. The heating rates are determined by taking images of a single ^{172}Yb^{+} ion's resonance fluorescence after a variable heating time and deducing the trapped ion's temperature from measuring its average oscillation amplitude. Assuming a power law for the heating rate versus ion-surface separation dependence, an exponent of -3.79±0.12 is measured.
We model electric field noise from fluctuating patch potentials on conducting surfaces by taking into account the finite geometry of the ion trap electrodes to gain insight into the origin of anomalous heating in ion traps. The scaling of anomalous heating rates with surface distance
d is obtained for several generic geometries of relevance to current ion trap designs, ranging from planar to spheroidal electrodes. The influence of patch size is studied both by solving Laplace's equation in terms of the appropriate Green's function as well as through an eigenfunction expansion. Scaling with surface distance is found to be highly dependent on the choice of geometry and the relative scale between the spatial extent of the electrode, the ion-electrode distance, and the patch size. Our model generally supports the ${d}^{\ensuremath{-}4}$ dependence currently found by most experiments and models, but also predicts geometry-driven deviations from this trend.