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

physics.ed-phSep 25, 2026

Artificial Neural Network Assisted Modelling of Tangent Galvanometer Measurements for the Determination of Horizontal Component of Earth's Magnetic Field

Authors: Saralasrita Mohanty, Sudakshina Prusty, Anshuman Pal, Pradipta Kumar Mishra

Organizations: School of Physical Sciences, National Institute of Science Education and Research, Jatni, Odisha 752050, India · Faculty of Engineering and Technology, Sri Sri University, Cuttack, Odisha 754006, India

Abstract

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.

Figures & tables

Subset	Nominal proportion	Purpose
Training	70%	Parameter optimization and model fitting
Validation	15%	Architecture selection and early stopping
Testing	15%	Held-out final model evaluation

Table 1: Dataset partitioning for ANN training, validation, and testing.

Feature	Role	Nature
Current, $I$	ANN input	Measured experimental variable
Average deflection, $\theta$	ANN input	Measured experimental variable
$\tan\theta$	ANN input	Derived from measured $\theta$
Induced field, $B_{i}$	ANN input	Derived physical field
No. of turns, $N$	ANN input	Experimental variable
$B_{H}$	ANN output	Experimentally calculated target

Table 2: Description and role of ANN input and output variables.

Coil	No. of observations	Mean current (mA)	Minimum current (mA)	Maximum current (mA)
50-turn	113	84.17080	12.32	353.70
500-turn	110	8.48073	1.26	22.35

Table 3: Summary of the retained experimental observations after quality-control filtering.

Figure 1: Architecture of the feed-forward artificial neural network for $B_{H}$ .

Activation	Hidden layers	Best epoch	Validation $R^{2}$	Test $R^{2}$	Test RMSE ( $\mu$ T)	Test MAE ( $\mu$ T)
Tanh	1	2500	0.96480	0.98640	0.64800	0.47446
Tanh	2	1063	0.97558	0.98776	0.61471	0.36821
Tanh	3	1996	0.97782	0.99053	0.54076	0.32940
Tanh	4	746	0.97333	0.98939	0.57228	0.25399
Tanh	5	1216	0.97703	0.98789	0.61159	0.29483
ReLU	1	2498	0.93270	0.95460	1.18398	0.87702

Table 4: Performance of the ANN architectures across activation functions and network depths.

Figure 2: Validation $R^{2}$ as a function of network depth for Tanh and ReLU activation functions.

Model	Feature representation	Test $R^{2}$	Test RMSE ( $\mu$ T)	Test MAE ( $\mu$ T)	Test MSE ( $\mu$ T 2 )
A	$I,\theta,N$	0.89220	1.82445	0.96491	3.32860
B	$I,\theta,\tan\theta,N$	0.98721	0.62834	0.46561	0.39482
C	$I,\theta,\tan\theta,B_{i},N$	0.99053	0.54076	0.32940	0.29242

Table 5: Test dataset performance of models A, B and C.

Figure 3: Test dataset performance of Models A, B, and C as physically derived input variables are progressively included.

Figure 4: Training and validation MSE during training of the final ANN. The dashed line marks the selected epoch (1996).

Dataset	$n$	$R^{2}$	RMSE ( $\mu$ T)	MAE ( $\mu$ T)	MSE ( $\mu$ T 2 )
Training	155	0.98564	0.68693	0.28522	0.47187
Validation	34	0.97782	0.89941	0.38595	0.80894
Testing	34	0.99053	0.54075	0.32940	0.29241

Table 6: Performance metrics for the combined training-validation and test dataset.

Figure 5: Regression performance of Model C for the training, validation, and testing subsets; the dashed line is the 45 ∘ reference.

Figure 6: Residual error versus observation frequency for the test dataset.

Coil	$N$	Mean current (mA)	Minimum current (mA)	Maximum current (mA)	$R^{2}$	RMSE ( $\mu$ T)	MAE ( $\mu$ T)
50-turn	113	84.17080	12.32	353.70	0.99678	0.30629	0.21591
500-turn	110	8.48073	1.26	22.35	0.98495	0.70073	0.44289

Table 7: Performance of the combined ANN model by coil configuration.

Figure 7: Residuals versus applied current for the held-out test set.

Method	Mean $B_{H}$ ( $\mu$ T)	MAE to $B_{H,\mathrm{ref}}$ ( $\mu$ T)	RMSE to $B_{H,\mathrm{ref}}$ ( $\mu$ T)	MAPE (%)
Experimental	37.38898	5.09088	5.92924	13.05354
ANN	37.32161	5.10273	5.92381	13.08393

Table 8: Comparison of experimental and ANN estimates with the adopted local geomagnetic horizontal reference.

Figure 8: Comparison of experimental and ANN-predicted value of $B_{H}$ for (a) 50-turn and (b) 500-turn coil. The blue dashed line denotes the adopted local geomagnetic horizontal reference of 39.0 $\mu$ T.

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Accurate reconstruction of magnetic fields in inaccessible regions is vital for many high-precision experiments in physics. Traditional methods, such as spherical harmonic expansion, often suffer from truncation errors that limit their precision. This study proposes an advanced Physics-Informed Neural Network (PINN) framework for high-precision 3D magnetic field mapping. Unlike conventional data-driven models, the proposed PINN integrates Maxwell's equations directly into the loss function, enforcing divergence-free and curl-free conditions across the entire domain. A key innovation is the inclusion of explicit physics-residual losses at measurement locations, ensuring rigorous physical consistency beyond random collocation sampling. Validation using simulated data achieves a reconstruction accuracy of

10^{-4}

, a tenfold improvement over existing PINN benchmarks. Furthermore, experimental validation using a custom coil assembly demonstrates robust reconstruction with sub-percent relative accuracy, reaching the

10^{-3}

level under ambient conditions. This AI-driven methodology provides a robust, high-precision solution for field monitoring and measurement in complex experimental environments where direct sensor placement is restricted.

Abstract

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