  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 $d^{-4}$ dependence currently found by
most experiments and models, but also predicts geometry-driven deviations from
this trend.
