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Artificial Neural Network Assisted Modelling of Tangent Galvanometer Measurements for the Determination of Horizontal Component of Earth's Magnetic Field
Saralasrita Mohanty, Sudakshina Prusty, Anshuman Pal, Pradipta Kumar Mishra
TL;DR
Conventional Tangent Galvanometer estimates of BH vary across observations because of measurement and environmental factors. An ANN predicts conventional Tangent Galvanometer estimates of BH from current, deflection angle, number of turns, tan θ, and the magnetic field at the coil centre, Bi. Model C achieved R² = 0.99053, RMSE = 0.54076 µT, and MAE = 0.32940 µT; mean experimental and ANN-predicted BH values were 37.38898 µT and 37.32161 µT.
Problem
Conventional Tangent Galvanometer estimates of BH vary across observations because of measurement and environmental factors.
Source: §1 · p. 1Method
An ANN predicts conventional Tangent Galvanometer estimates of BH from current, deflection angle, number of turns, tan θ, and the magnetic field at the coil centre, Bi.
Source: §1 · p. 2, §2.1 · p. 2Results
Model C achieved R² = 0.99053, RMSE = 0.54076 µT, and MAE = 0.32940 µT; mean experimental and ANN-predicted BH values were 37.38898 µT and 37.32161 µT.
Source: §6 · p. 10Takeaways & Limitations
ANN modeling can be used with this undergraduate experiment to study experimental variations and nonlinear relationships while retaining the connection to underlying physics.
Source: §6 · p. 10Further work with larger datasets and repeated experiments is needed to check reliability; the ANN is an empirical modeling layer using features tied to the conventional TG relationship, not an independent derivation of geomagnetic field.
Source: §2.1 · p. 2, §6 · p. 10
Abstract
from arXiv · showhide
The Tangent Galvanometer (TG) is a standard undergraduate laboratory experiment for estimating the horizontal component of Earth's magnetic field (BH) by measuring the angle of deflection of a magnetic needle corresponding to the current flowing through a circular coil. In this study, an artificial neural network (ANN) is used as a complementary data-driven model to predict the value of BH. A dataset comprising 225 observations obtained using 50-turn and 500-turn coils was used for developing the ANN model. After quality control, 223 observations were retained and divided into training (70%), validation (15%), and testing (15%) subsets. The model was optimized using a feed-forward neural network with Tanh activation. Three different models (Models A, B, and C) with different input variables were compared for optimum performance. Model C used five input variables: current, deflection angle, tan(theta), magnetic field produced by the coil, and number of turns. The addition of tan(theta) produced a substantial improvement in prediction performance, which was further improved by including the magnetic field produced by the coil. Model C gave the best test performance, with R2 = 0.99053, RMSE = 0.54076 microT, and MAE = 0.32940 microT. The experimental and ANN-predicted values of BH were also compared with an adopted local geomagnetic reference value of 39.0 microT. The mean experimental and ANN-predicted values were 37.38898 microT and 37.32161 microT, respectively. The results demonstrate the usefulness of ANN as a complementary tool for analyzing experimental variability and nonlinear relationships in an undergraduate physics laboratory experiment.
1 Introduction
The Tangent Galvanometer demonstrates how a current-carrying coil interacts with Earth’s horizontal magnetic field, with needle deflection illustrating the tangent law. This study extends its conventional analysis with ANN modeling while retaining the physical meaning of its variables.
1 Introduction
- The Tangent Galvanometer’s needle deflection illustrates the tangent law when the coil field is approximately perpendicular to Earth’s horizontal geomagnetic field.ANNs are presented as flexible nonlinear regression models for relating measured or derived variables to an experimental target.Source: §1 · p. 1
- Instrument resolution, current fluctuations, alignment, pivot friction, parallax, coil tolerances, and environmental disturbances contribute to observation-to-observation variation in calculated BH.Conventional analysis obtains BH from measured current and deflection through the tangent-law relation.Source: §1 · p. 1
- The study uses an ANN as a complementary data-driven model, comparing basic measurements with physically derived inputs and examining residuals, coil configurations, and simpler regression baselines.Predictions are compared with conventional estimates and an adopted local geomagnetic reference.Source: §1 · p. 2
2 Tangent Galvanometer: Physical Principle
The coil’s center field is treated as a derived physical feature, while the equilibrium deflection follows the tangent law when the coil field is perpendicular to Earth’s horizontal field.
2 Tangent Galvanometer: Physical Principle
- For a circular coil, its center magnetic field is determined by current, number of turns, and coil radius; the study uses Bi as a derived physical feature.The passage identifies µ0 as the permeability of free space.Source: §2.1 · p. 2
- The tangent-law relation applies when the coil field is perpendicular to Earth’s horizontal magnetic field BH and the needle reaches equilibrium deflection θ.The ANN target is the conventional TG estimate, and tan θ and Bi are inputs derived from the same experiment; the model is therefore an empirical prediction and error-modelling layer, not an independent derivation of geomagnetic field.Source: §2.1 · p. 2
3 Experimental Procedure and Dataset
Measurements from 50-turn and 500-turn coils supplied current, deflection, and derived BH observations; the retained dataset was split for model development and testing.
3.1 Experimental arrangement and measurements
- The experiment used 50-turn and 500-turn coils oriented in the magnetic meridian, varied current systematically, and recorded needle deflection at stable equilibrium.Multiple readings were averaged for each selected current, and current direction was reversed when needed to reduce systematic effects.Source: §3.1 · p. 2
- Instrument resolution, current fluctuations, alignment, pivot friction, parallax, coil tolerances, and environmental disturbances contribute to variation in calculated BH across observations.Each observation paired an experimental BH target with measured and derived quantities for ANN modeling.Source: §3.1 · pp. 2–3
3.2 Data splitting and preprocessing
- 223 observations were divided into training, validation, and test subsets using a 70%-15%-15% split stratified by coil turns.Inputs and target were standardized using parameters fitted only on the training subset, then applied to validation and testing data.Source: §3.2 · p. 3, Table 1 · p. 3
4 ANN Methodology
The ANN predicts conventional TG-derived BH from measured and derived experimental variables, using a combined dataset, validation-based architecture selection, and standard error metrics.
4.1 Input and target variables
- The five model inputs were current I, average deflection θ, tan θ, induced field Bi, and turns N; experimentally calculated BH was the target.The derived features tan θ and Bi are not additional independent measurements.Source: §4.1 · p. 3, Table 2 · p. 3
4.2 Experimental dataset and quality control
- Quality control removed two of 225 observations, retaining 223: 113 from the 50-turn coil and 110 from the 500-turn coil.Observations were excluded for θ outside 0°–90° or calculated BH ≥100 µT; retained current ranges differed between the two coil configurations.Source: §4.2 · pp. 3–4, Table 3 · p. 4
4.3 ANN architecture and training
- The selected feed-forward ANN architecture was 5→32→32→32→1, with 2,337 trainable parameters, selected by validation R² and trained with early stopping.Tanh and ReLU were compared across one to five hidden layers; the test set was not used for model selection.Source: §4.3 · p. 4, Figure 1 · p. 4
4.4 Evaluation metrics
- ANN performance was evaluated using R², MSE, RMSE, MAE, and MAPE, with the residual defined as the difference between experimental target and prediction.The same MAE, RMSE, and MAPE definitions were applied to experimental estimates, which were compared with a local 39.0 µT geomagnetic reference.Source: §4.4 · pp. 4–5
5 Results and Discussions
Model C had the best test error metrics among the input variants, and its R2 was close to 1 across training, validation, and test sets. The analyses also show that residuals remain and that performance differs across coil configurations.
5.1 Network-depth and activation-function analysis
- Tanh with three hidden layers gave optimum performance, while increasing depth did not automatically improve generalization.The analysis compared Tanh and ReLU networks with up to five hidden layers.Source: §5.1 · p. 5, Table 4 · p. 5, Figure 2 · p. 6
5.2 Influence of input variables on model performance
- Adding tan θ in Model B substantially changed the input representation, while adding Bi in Model C further reduced test error; Model C had the optimum test error metrics.The network architecture, data partition, and training procedure were held fixed across the three models.Source: Table 5 · p. 6, §5.2 · pp. 6–7, Figure 3 · p. 6
5.3 Training and validation behavior of the final model
- Training and validation MSE fell rapidly early and then stabilized; the final parameters came from epoch 1996, when validation performance was best.Test MSE was calculated only after the final model had been selected.Source: §5.3 · p. 7, Figure 4 · p. 7
5.4 Evaluation of performance metrics
- Model C’s R2 was close to 1 across training, validation, and test sets, with test data excluded from fitting and architecture selection.Table 6 summarizes the final model’s performance metrics.Source: Table 6 · p. 7, §5.4 · p. 7, Figure 5 · p. 8
5.5 Residual Error analysis on test dataset
- Most test residuals were concentrated near zero, but a few larger deviations remained, particularly on the positive side, and measurement variation persisted.Figure 6 presents residual error against observation frequency.Source: §5.5 · p. 8, Figure 6 · p. 8
5.6 Performance of Model C for different coil configuration
- The 500-turn configuration’s larger RMSE and MAE were attributed to its substantially lower-current operating regime, while possible other contributors were not independently quantified.Figure 7 shows residuals against applied current for the held-out test set.Source: §5.6 · p. 8, Table 7 · p. 9, Figure 7 · p. 9
5.7 Comparison of Experimental and ANN predicted value of BH
- Across 223 observations, mean experimental and ANN-predicted BH were 37.38898 µT and 37.32161 µT, respectively, and their error metrics against the local reference were similar.The comparison includes an adopted local geomagnetic reference of 39.0 µT.Source: Table 8 · p. 9, §5.7 · p. 9, Figure 8 · p. 10
6 Conclusion
The ANN complements conventional Tangent Galvanometer analysis, with the five-input Model C producing the reported optimum error metrics. Its predictions closely represent the experimental estimates relative to the adopted local geomagnetic reference; larger datasets and repeated experiments are needed to check reliability.
6 Conclusion
- Model C, using all five inputs, yielded R2 = 0.99053, RMSE = 0.54076 µT, and MAE = 0.32940 µT.Adding tan θ produced the largest performance change, with further improvement after including Bi.Source: §6 · p. 10
- 37.38898 µT experimental and 37.32161 µT ANN-predicted mean BH values showed very similar deviations from the adopted 39.0 µT local reference.The authors interpret this similarity as close representation of the experimental measurements rather than a substantial shift in estimated BH.Source: §6 · p. 10
- ANN modelling can study experimental variations and nonlinear relationships while retaining the connection to underlying physics.The authors call for larger datasets and repeated experiments to check reliability; adding environmental and instrument factors may further improve prediction.Source: §6 · p. 10