Design and Testable Prediction of a Low-Cost Magnetic Force Measurement Scheme Based on Static Equilibrium

Author: Zhang Suhang, Luoyang, Henan

Abstract

Magnetic force is a non-contact force. Conventional measurements generally rely on magnetic field sensors and precision force measuring devices. Under low-cost conditions, it is difficult to carry out mutual verification from four dimensions: attraction, repulsion, frictionless equilibrium and electromagnetic comparison. Based on the classical static equilibrium principle, this paper designs four mutually verifiable low-cost magnetic force measurement schemes: vertical suspension weight method for like-pole repulsion, horizontal spring/rubber band equilibrium method for opposite-pole attraction, suspension angle decomposition method, and electromagnetic-permanent magnet comparison method under identical geometry. The four schemes convert magnetic force into gravity, elastic force, tension component of suspension thread and empirical relation of electromagnetic force respectively, forming a self-consistent and reproducible experimental matrix. On this basis, magnetic charge approximation, magnetic dipole far-field approximation and empirical power-law model are introduced to provide testable predictions for the magnetic force–spacing relation: the power-law exponent n can be determined by the slope of the double-logarithmic plot. This paper presents conceptual design and feasibility analysis without physical measurement. It aims to provide a scheme framework for low-cost magnetic force experiments, basic physics demonstrations and subsequent experimental verification.

Keywords: magnetic force measurement; static equilibrium; magnetic repulsion; magnetic attraction; experimental design; testable prediction

1. Introduction

Quantitative measurement of magnetic force usually requires special force measuring devices or magnetic field sensors. Under ordinary conditions, it is difficult to conduct low-cost mutual verification from multiple directions including magnetic attraction, magnetic repulsion, frictionless equilibrium and electromagnetic comparison. In existing basic physics experiments, magnetic force demonstrations mostly focus on qualitative phenomena. There lacks a set of simple, reproducible experimental schemes that take both magnetic attraction and repulsion into account and can be linked to theoretical power-law models.

This paper constructs four independent and mutually verifiable magnetic force measurement methods, covering like-pole repulsion measurement, opposite-pole attraction measurement, frictionless high-precision measurement and electromagnetic calibration. The whole setup is mainly composed of common components such as graduated cylinders, thin threads, rubber bands, paper clips and rulers, suitable for principle demonstration and semi-quantitative testing. This paper does not claim to derive a new magnetic force formula; instead, it provides testable model assumptions and experimental criteria for subsequent researchers to build physical setups, perform actual measurement and error correction.

2. Theoretical Models and Testable Predictions

There is no universal “all-purpose formula” for magnetic force comparable to Coulomb’s law, but testable forms exist under different approximations.

2.1 Magnetic Charge Approximation

Treat pole faces as magnetic charge q_m:

F=\frac{\mu_0}{4\pi}\frac{q_{m1}q_{m2}}{r^2}

This formula has the same form as Coulomb’s law, suitable for teaching models and semi-quantitative estimation near the pole face. Magnetic monopoles do not exist, so this model is only valid under specific near-field approximation.

2.2 Magnetic Dipole Far-Field Approximation

Real permanent magnets can be regarded as magnetic dipoles in the far field. For two collinear axial dipoles, the approximate force magnitude reads:

F\approx \frac{3\mu_0}{2\pi}\frac{m_1m_2}{r^4}

The magnetic field of a dipole decays as 1/r^3. Force is the gradient of magnetic field, so the force tends to follow 1/r^4 in the far field.

2.3 Empirical Power Law

For magnets at finite spacing, an empirical formula is commonly adopted:

F=\frac{C}{(d+d_0)^n},\qquad n\approx 2\sim 4

where d is the spacing between pole faces, d_0 is the equivalent zero-offset correction, C depends on magnet strength, geometry and medium, and n is determined by distance range and geometry. The near-field pole-face approximation gives n\approx 2, while the far-field dipole approximation yields n\approx 4.

2.4 Testable Criterion

Take double logarithm of the measured F(d):

\log F=\log C-n\log(d+d_0)

If the curve is approximately linear within a certain interval, the slope equals -n. All four schemes in this paper aim to measure the F(d) or relative force–spacing trend. The value of n is treated as a quantity to be tested rather than a preset constant. This serves as the theoretical prediction and experimental interface of this work.

3. Design of Four Magnetic Force Measurement Schemes

Scheme 1: Vertical Suspension Weight Method for Like-Pole Repulsion (Repulsion Measurement)

Principle

Two coaxial disc permanent magnets are placed with like poles facing each other. The lower magnet is fixed at the bottom of a graduated cylinder, and the upper magnet is centered with a lightweight foam ring and suspended by magnetic repulsion.

Vertical force equilibrium:

F_{\text{rep}}(d)=W

where W is the total weight of the upper magnet and added weights, and d is the equilibrium spacing. Change the weight by adding paper clips, record the equilibrium position from the cylinder scale to obtain the weight–spacing curve.

Procedures

1. Fix the lower magnet at the bottom of the graduated cylinder with like pole upward; fit the upper magnet with a lightweight foam ring and place it inside the cylinder.

2. Place the graduated cylinder vertically and calibrate with a straightedge or spirit level.

3. Gently add paper clips one by one on top of the upper magnet. Wait for stabilization and record the position of the upper magnet to calculate equilibrium spacing d.

4. Record “number of paper clips — equilibrium spacing” and convert to relative force W.

5. Replace with magnets of the same model or change pole faces for repeated comparison.

Features

1. Simple and intuitive setup; the levitation phenomenon of magnetic repulsion has obvious visual effect.
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2. Magnetic force balanced by self-weight, eliminating additional error caused by spring deformation.
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3. Convenient to compare repulsion strength under different magnets and spacing.

Error Sources and Improvements

Lateral displacement and overturning of magnets, air disturbance, non-ideal dipole characteristic of magnets.
Improvements: use lightweight centered foam ring; take the median value of equilibrium range after gentle tapping; use wind shield; average repeated readings; use relative force–spacing curve instead of taking single height reading as absolute magnetic force.

Scheme 2: Horizontal Spring/Rubber Band Equilibrium Method for Opposite-Pole Attraction (Attraction Measurement)

Principle

The graduated cylinder is fixed horizontally. Magnet A is fixed at one end, and magnet B fitted with foam ring is placed inside the cylinder with opposite poles facing each other. Magnetic attraction stretches the rubber band or spring connecting magnet B to the cylinder opening. Neglecting minor sliding friction:

F_{\text{att}}(d)\approx kx

where x is the elongation of rubber band or spring, and k needs pre-calibration. Read the position of magnet B to obtain spacing d, so as to acquire the attraction–spacing trend.

Procedures

1. Fix the graduated cylinder horizontally, mount magnet A at one end.
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2. Fit magnet B with lightweight foam ring and place it inside the cylinder, with opposite pole facing magnet A.
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3. Connect magnet B to the cylinder opening with a rubber band. Calibrate the rubber band in advance: hang known numbers of paper clips to measure elongation and obtain relative value of k.
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4. Adjust the fixed position of magnet A or replace rubber band, record equilibrium position and elongation of magnet B.
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5. Plot the elongation–spacing curve.

Features

1. Horizontal layout eliminates gravity interference on attraction measurement.
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2. Direct measurement of magnetic attraction; simple structure and easy assembly.

Error Sources and Improvements

Sliding friction between magnet and cylinder wall; nonlinearity and aging of rubber band; attitude deflection of magnet.
Improvements: lightweight closely-fitted foam ring; pre-calibrate rubber band; take median value of repeated equilibrium readings; only for trend and semi-quantitative comparison, not for high-precision claims.

Scheme 3: Suspension Angle Decomposition Method for Opposite-Pole Attraction (Frictionless Attraction Measurement)

Principle

One magnet is fixed, the other is suspended by thin thread. Opposite poles attract each other. The suspended magnet is subjected to horizontal magnetic force and gravity. At equilibrium, the suspension thread deflects by angle \theta:

F_{\text{att}}=mg\tan\theta

Measure the spacing d between pole faces simultaneously to obtain F(d). This scheme has no solid contact friction and possesses the most concise theoretical relation.

Procedures

1. Mount the fixed magnet on a support with vertical pole face.
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2. Suspend the second magnet with thin thread; pre-twist release by hanging weight until static state.
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3. Move the fixed magnet so opposite poles face each other, causing deflection of the suspended magnet.
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4. Measure horizontal displacement and thread length with ruler to calculate \theta, or measure angle from slow-motion photos taken by mobile phone.
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5. Measure pole-face spacing d at the same time, record \theta-d and convert to F=mg\tan\theta.
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6. Plot F-d curve and generate double-logarithmic plot to read slope.

Features

1. Lowest friction impact and highest theoretical rigor among the four schemes.
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2. No solid contact friction; magnetic force converted into angle measurement. The same magnets can be reused with Scheme 1 and 2 for attraction/repulsion trend comparison.

Error Sources and Improvements

Thread torsion, air flow disturbance, persistent small swing of magnet, spacing measurement error.
Improvements: release residual torsion before test; use carton for wind shielding; average repeated angle readings; measure d against fixed reference plane; attach small spirit level to magnet to control attitude.

Scheme 4: Electromagnetic-Permanent Magnet Comparison Method under Identical Geometry

Principle

An electromagnet generates adjustable magnetic force with fixed air gap. Use Scheme 2 or 3 to measure electromagnetic force F_{\text{em}} under different current I, establishing empirical relation:

F_{\text{em}}=f(I,d)

Then replace with a permanent magnet of identical geometry and spacing, measure F_{\text{pm}}(d), and compare force decay trend versus spacing. This scheme does not claim that current directly calibrates magnetic force, but builds empirical comparison under identical geometry.

Procedures

1. Fix the electromagnet and keep air gap d constant.
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2. Measure electromagnetic force under different current by suspension method or horizontal rubber band method.
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3. Record current–force curve, paying attention to core saturation, heating and remanence.
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4. Replace with permanent magnet of the same size, maintain identical geometry and spacing, measure permanent magnet force.
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5. Compare force trends against d for the two cases, without assuming unconditional absolute equivalence.

Features

1. Electromagnetic force can be adjusted continuously.
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2. Comparison between electromagnet and permanent magnet under identical geometry, opening the comparative measurement path for permanent magnetism and electromagnetism.

Error Sources and Improvements

Core saturation, remanence, heat generation, air gap measurement error.

Improvements: multi-point measurement with increasing current; limit power-on duration; fix air gap; record temperature; only for trend comparison; do not treat simple linear relation as universal law.

4. Complementary Logic of the Four Schemes

表格

Scheme Measured Quantity Equilibrium Mechanism Friction Influence Main Error Positioning 

Scheme 1 Magnetic repulsion Gravity equilibrium Moderate (cylinder wall) Lateral instability, friction Repulsion trend measurement 

Scheme 2 Magnetic attraction Elastic force equilibrium Moderate (sliding) Rubber band nonlinearity Horizontal attraction semi-quantitative measurement 

Scheme 3 Magnetic attraction Tension component of suspension thread Low Torsion, air flow Low-friction attraction measurement 

Scheme 4 Electromagnetic force / permanent magnet comparison Empirical relation of electromagnetic force Depends on adopted force measuring method Saturation, remanence, heating Electromagnetic–permanent magnet comparison 

The four schemes are not listed in isolation. Scheme 1 measures magnetic repulsion, Scheme 2 performs horizontal static measurement of magnetic attraction, Scheme 3 provides low-friction measurement of magnetic attraction, and Scheme 4 establishes geometric comparison between electromagnetic force and permanent magnet force. Combined, they serve one testable objective: to measure F(d) trend and estimate power-law exponent n.

5. Research Value

1. Low-cost experimental matrix: Magnetic force trend measurement can be realized with ordinary components such as graduated cylinders, thin threads and rubber bands. It is suitable for physics demonstrations in middle schools and universities, and principle exploration by independent researchers without access to professional equipment.

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2. Multi-dimensional mutual verification: Covering repulsion, attraction, frictionless equilibrium and electromagnetic comparison, it compensates for the limitation of single measurement method.

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3. Clear theory-experiment interface: Convert magnetic charge approximation, dipole far-field model and empirical power law into double-logarithmic slope criterion. The scheme is more than qualitative demonstration and carries testable properties.

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4. Reproducible design: Procedures, error sources and improvement directions are provided, facilitating subsequent researchers to build physical setups, carry out measurement and error correction.

6. Research Statement

All experimental structures in this paper are conceptual design and feasibility analysis. No physical measurement has been performed and no experimental data are fabricated. The author is an independent researcher without access to a professional laboratory. This paper aims to establish a self-consistent, reproducible and falsifiable framework for magnetic force measurement. All cited theoretical models are existing approximations; this paper does not claim to propose a new magnetic force formula. Follow-up work may prioritize verification of Scheme 3, then use Scheme 1 and 2 for attraction/repulsion trend comparison, and Scheme 4 for electromagnetic-permanent magnet comparison under identical geometry.

Safety Notice: Strong magnets carry risks of pinching injury, ejection and attraction of ferrous objects. Glass graduated cylinders are fragile and may break upon magnet collision. Laser or intense light shall not be directed into eyes. Electromagnets shall not be powered for excessive duration to avoid overheating.

7. Conclusion

Based on the principle of static equilibrium, this paper designs four low-cost magnetic force measurement schemes, forming an experimental matrix covering magnetic repulsion, magnetic attraction, frictionless angle measurement and electromagnetic-permanent magnet comparison. Built upon common components including graduated cylinders, the four schemes convert magnetic force into gravity, elastic force, tension component of suspension thread and empirical relation of electromagnetic force. Magnetic charge approximation, dipole far-field approximation and empirical power law are introduced as testable predictions. This paper presents conceptual design and feasibility analysis without claiming high-precision experimental conclusions. Its value lies in providing a reproducible, falsifiable and low-threshold framework of magnetic force experiments for subsequent measurement verification and teaching demonstration.

References

[1] Zhao Kaihua, Chen Ximou. Electromagnetics[M]. Higher Education Press.

[2] Griffiths D J. Introduction to Electrodynamics[M]. Cambridge University Press.

[3] Feynman R P, Leighton R B, Sands M. The Feynman Lectures on Physics, Vol. II[M]. Addison-Wesley.

[4] National Metrology Specification. Evaluation and Expression of Uncertainty in Measurement (JJF 1059)[S].

[5] Chapters on static equilibrium, spring calibration and error analysis in general physics experiment textbooks[M].

[6] Chapters on empirical magnetic force formula and air-gap magnetic force in engineering magnetism textbooks[M].