Build a Green function with a declared convention

Here choose the positive operator minus the Laplacian, and define its Dirichlet Green function by a unit delta source. In free three-dimensional space the corresponding kernel is positive one over four pi times separation. Applying the operator produces a unit delta, not four pi times a delta. The electrostatic potential then includes an additional factor of inverse permittivity. The kernel has inverse-length units, while its convolution with charge density has potential units.

A finite-domain kernel adds a harmonic correction that sets the kernel to zero on the boundary. For a grounded plane, subtracting the reflected free-space kernel does exactly this. That correction is not an optional refinement: it encodes the electrodes. Some references instead define the kernel for the positive Laplacian and use a negative free-space kernel. Their representation formula changes signs consistently. Always compare the defining differential equation before borrowing a formula.

Apply Green’s second identity to the potential and the kernel in the source coordinate. The delta selects the potential at the observation point, while Poisson’s equation supplies the volume charge integral. Because the Dirichlet kernel vanishes on the boundary, the term involving the unknown normal derivative of the potential disappears. The remaining boundary integral involves the known boundary potential and the normal derivative of the kernel, with a minus sign in our convention.

If every boundary is grounded, that remaining term vanishes too. If there is no volume charge but the boundary is biased, the boundary term alone creates the solution. Omitting it would falsely predict zero potential everywhere. Symmetry of the Dirichlet kernel under exchanging its arguments follows from the same identity and homogeneous boundary conditions. This reciprocity statement relies on the self-adjoint operator and appropriate domain, rather than on any generic pair of response functions.

A closed-domain Neumann Laplacian annihilates constants. Consequently it cannot possess an ordinary inverse on all sources. One consistent generalized kernel satisfies minus its Laplacian equal to delta minus reciprocal domain volume, has zero normal derivative, and has zero volume mean. Subtracting the uniform term removes the incompatible constant source. Another convention keeps a pure delta and uses a nonzero constant boundary derivative. These conventions must not be combined with each other’s formulas.

Use the field lab only to test linear scaling of a source response. Reset and set q=0.5 and rotation to 0, then Reset and set q=1 and rotation to 0. Predict that the numerical Fx readout at (1,0) increases from 0.5/1.2≈0.417 to 1/1.2≈0.833, while Fy stays zero. The mathematical field doubles; visible arrow lengths use a compressed scale. This supports the superposition step behind a kernel integral. The softened planar field has neither a delta normalization nor conducting boundaries, so it cannot validate a Green function’s normalization, boundary term, or Neumann mean constraint.