Magnetization of Linear Arrays of Two Ferromagnetic Spheres in a Uniform Magnetic Field
Bibliographic record
Abstract
The main objective of this paper is to explore the exact analytical solutions of the magnetization of linear arrays of two ideal ferromagnetic spheres in the presence of external magnetic fields. First, the total scalar magnetic potential outside the spheres, related to the magnetic field intensity, which satisfies the Laplace equation, is obtained by the superposition of the potentials due to all spheres and the potential corresponding to the external field. The translational addition theorems for scalar Laplacian functions in spherical coordinates are then used to translate the two coordinates systems into one coordinates system. Then, the exact boundary conditions were used to solve the magnetic field quantities outside the system. On the other hand, the scalar magnetic potential inside each sphere, related to the magnetic flux density, also satisfies the Laplace equation, which is solved by imposing the boundary conditions known from the solution of the outside quantities. Finally, the expressions derived are used to generate numerical results of controllable accuracy for various field quantities.
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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 | 0.001 |
| 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) | 0.002 | 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".