designs can be accomplished through an analytical description of the electric field density based on conformal mapping. In this way, a two-dimensional approximation to coplanar waveguide and capacitor designs produces values of the participation as a function of depth from the top metallization layer as well as the volume participation within a given thickness from this surface by reducing the problem to a surface integration over the region of interest. These quantities are compared to finite element method numerical solutions, which validate the values at large distances from the coplanar metallization but diverge near the edges of the metallization features due to the singular nature of the electric fields. A simple approximation to the electric field energy at shallow depths (relative to the waveguide width) is also presented that closely replicates the numerical results based on conformal mapping. These techniques are applied to the calculation of surface participation within a transmon qubit design, where the effects due to shunting capacitors can be easily integrated with those associated with metallization comprising the local environment of the qubit junction.