Current Loop Off Axis Field Approximations With Excellent Accuracy and Low Computational Cost
Bibliographic record
Abstract
The current loop is a fundamental building block of cylindrically symmetric magnetic calculations. However, the off-axis magnetic field involves the subtraction of elliptic integrals of the first and second kind, which is computationally expensive, hard to manipulate in equations, and difficult to visualize. By conducting a different binomial series expansion on the original loop integral, a series solution is created, which can be simplified to a set of approximation functions with useful characteristics: exactly correct along the axis and at distance, while in the current loop itself the relative error is limited, computationally simple, highly accurate, and easy to visualize for behavior or symmetry. For the radial field, the first and second orders fit where the parameters are optimized to minimize the relative peak error. Note the symmetry that occurs by expressing as function of the scaled expansion term <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula> , which ranges from 0 to 1, allowing maximum relative errors of 0.025 for first order and 2.9E-4 for second order. For the axial field first order, due to the subtraction of terms, the accuracy is only modest but the second order has a 9E-4 maximum relative error, with zero error at the loop, on the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z =0$ </tex-math></inline-formula> loop plane. The axial field relative error stays low within the loop, but the outside stays low for any h until it falls to 1% of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z =0$ </tex-math></inline-formula> plane value, then it loses accuracy as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> is near where the field direction reverses sign due to slight differences in the zero crossing predicated coordinate, increasing again in accuracy as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> moves further from the reveal. Higher order approximations for both radial and axial add more <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula> terms with increasing accuracy—for radial, the third order is 1.8E-5, fourth order is 2.3E-6, and fifth order is 4.9E-7, while axial goes as third 4.6E-5, fourth 6E-6, and fifth 1.3E-6. Relative error plots in 3-D space are presented for all orders of approximations. The simplicity of these functions suggests new ways combining loops to optimize such things as field uniformity.
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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) | 0.000 | 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".