Integral Equations of Electrostatics in Multi‐Region Scenarios with Free‐Space Green's Functions
Bibliographic record
Abstract
After reading this Appendix you should be able to:• Derive integral equations of electrostatics for multi-region metal dielectric objects situated in free space. J.1 Equivalence Principle for the External Electrostatic Field in Layered MediaIn this section we derive integral equations for the electrostatic fields in layered media where the layers are handled as finite size dielectric regions and free-space Green's function is utilized.Common application of the described integral equation formulations is in the numerical schemes for capacitance and conductance extraction in 2D and 3D systems of conductors (e.g.transmission lines) embedded in layered substrates.The formulation generalizes derivation presented in Bladel (2007) for the case of metal/dielectric multi-region problem depicted in Figure J.4.For simplicity of derivations we consider a single volume V 0 in free space.The statement of the boundary value problem is the following.The scalar potential 𝜙 must satisfy homogeneous Laplace equation 1 4𝜋𝜖 0 |r-r ′ | of homogeneous medium with permittivity 𝜖 0 has the meaning of the potential at observation point r produced by the point charge Q = 1 [Coulomb] situated Theory and Computation of Electromagnetic Fields in Layered Media, First Edition.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".