Integral Equations of Electrostatics in Multi‐Region Scenarios with Free‐Space Green's Functions
Bibliographic record
Abstract
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 must satisfy homogeneous Laplace equationat all points except for the ones located on the dielectric interfaces.At the dielectric interfaces potential satisfies the following boundary conditionsIn (J.2) 0 ε-, 0 ε+ denote permittivities at the points located immediately inside and outside dielectric interface S, respectively considering direction of chosen normal at S. If surface S is that of a PEC object, potential assumes constant value V on it, i.e. (r) = V| r∈S .Green's function G(r, r ′ ) = 1 4 0 |r-r ′ | of homogeneous medium with permittivity 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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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 teacher head, 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".