In a finite difference (FD) based electromagnetic formulation based on a Yee cell grid, one can define electric current source excitations ($J$) on the $E$ field grid points. At a distance, the fields generated from these excitations (in free-space) closely match with those of an analytical Hertzian dipole (which is infinitesimally thin). A Hertzian dipole has a constant current $I$ over its small length $l$, but since it is infinitesimally thin, its effective current area density $J = \frac{I}{dxdy} = \infty$, so the fields are singular at the origin and do not match up with those of the "real" current in a FD formulation (which is kind of like a cube of current rather than a line and thus has finite $J$ and finite $E$ everywhere including the near-field) close to the source. If I have multiple of these sources in my system at different locations, I can compute the power radiated from each individual one by integrating the quantity $J\cdot E$ over a small volume enclosing each source.
In an integral equation (IE) formulation, instead of directly specifying current excitations, one just specifies incident fields and the corresponding currents over dielectric/conducting surfaces and scattered fields are computed. How could I achieve the same thing as a current excitation with finite dimensions with an IE formulation (like with a FD method)?
I guess I could use the closed-form fields from an analytical Hertzian dipole as an incident wave source and calculate scattered fields, however, this would not have the same result as a finite current "cube" in an FD formulation. Is there any straightforward way to define electric current excitations with finite current density J (or at least surface current density) in a surface integral equation formulation? Further, how would you go about calculating the incident power from each source when using an IE method?
