A robust field-only boundary integral formulation of electromagnetics is derived without the use of surface currents that appear in the Stratton–Chu formulation. For scattering by a perfect electrical conductor (PEC), the components of the electric field are obtained directly from surface integral equation solutions of three scalar Helmholtz equations for the field components. The divergence-free condition is enforced via a boundary condition on the normal component of the field and its normal derivative. Field values and their normal derivatives at the surface of the PEC are obtained directly from surface integral equations that do not contain divergent kernels. Consequently, high-order elements with fewer degrees of freedom can be used to represent surface features to a higher precision than the traditional planar elements. This theoretical framework is illustrated with numerical examples that provide further physical insight into the role of the surface curvature in scattering problems.

One of the most enduring, broadly applicable and widely used theoretical results of electrokinetic theory is the Smoluchowski expression for the electrophoretic mobility. It is a limiting form that holds for any solid particle of arbitrary shape in an electrolyte of any composition provided the thickness of the electrical double layer is "infinitely" thin compared to the particle size and the particle has uniform surface potential. The familiar derivation of this result that is a simplified version of the original Smoluchowski analysis in 1903, considers the motion of the electrolyte adjacent to a planar surface. The theory is deceptively simple but as a result much of the interesting physics and characteristic hydrodynamic behavior around the particle have been obscured. This paper provides a derivation of this key theoretical result by starting from Smoluchowski’s original 1903 analysis but brings out overlooked details of the hydrodynamic features near and far from the particle that have not been canvassed in detail. The objective is to draw together all the key physical features of the electrophoretic problem in the thin double layer regime to provide an accessible and complete exposition of this important result in colloid science.