arxiv: 2605.03293 · v1 · submitted 2026-05-05 · ⚛️ physics.ed-ph

Recognition: 2 theorem links

· Lean Theorem

A smartphone-based simple method for determination of the free space permeability

Sanjoy Kumar Pal , Soumen Sarkar , Pradipta Panchadhyayee

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:36 UTC · model grok-4.3

classification ⚛️ physics.ed-ph

keywords smartphonefree space permeabilityterminal velocityconducting pipemagnetic sensorfalling magnetphysics education

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The pith

Free-space permeability is computed from the terminal velocity of a magnet falling inside a conducting pipe measured with a smartphone sensor and video playback.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a laboratory procedure that extracts the permeability of free space directly from the constant speed reached by a magnet dropped through a metal pipe. Timing and position data come from the phone's built-in magnetometer together with ordinary video recording of the fall. The authors treat the result as accurate and precise enough to replace more elaborate apparatus in teaching settings. A reader would care because the method converts a familiar classroom demonstration into a quantitative determination of a basic constant using only everyday items.

Core claim

The permeability of free space is obtained by substituting the measured terminal velocity of the falling magnet into the standard electromagnetic-drag expression for motion inside a conducting tube; the velocity itself is extracted from the time-varying magnetic field signal recorded by the smartphone sensor while the video confirms the interval of constant speed.

What carries the argument

The terminal-velocity formula for a magnet in a conducting pipe, which equates gravitational force to the opposing force from induced eddy currents whose strength depends on the permeability of free space.

If this is right

The procedure supplies a numerical value of permeability that matches the accepted constant when the terminal velocity is recorded accurately.
No auxiliary instruments beyond a smartphone and a conducting tube are required to obtain the result.
The same data set yields both the velocity and the permeability in a single run.
The method can be performed in ordinary classrooms or at home without access to research-grade magnetometers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same smartphone recording could be re-analyzed to extract pipe conductivity if permeability is taken as known.
Variants of the drop could test how terminal velocity scales with magnet strength or pipe diameter as a student exercise.
The approach may extend to other electromagnetic demonstrations where a phone sensor replaces a dedicated probe.

Load-bearing premise

The theoretical expression for terminal velocity isolates permeability without meaningful interference from pipe conductivity details, magnet orientation, or errors in the smartphone sensor reading.

What would settle it

Repeating the drop with pipes of different conductivity or wall thickness and finding that the computed permeability changes by more than experimental uncertainty would show the formula does not cleanly isolate the free-space value.

read the original abstract

A simple and novel method is designed to determine the free space permeability. This value is computed from the expression of the terminal velocity of a magnet falling through a conducting pipe using the magnetic sensor of a smartphone and a video player. This method deserves its importance because of the accuracy and precision of the results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a smartphone lab exercise for measuring μ0 that adapts a standard eddy-current setup but does not cleanly isolate the constant.

read the letter

The paper walks through dropping a magnet down a conducting pipe and using a phone's magnetometer plus video timing to get the terminal velocity, then solving for the free-space permeability from the force-balance equation. The setup is cheap and uses tools most students already have, which is the practical angle here. It could work as a quick classroom or home demo to show magnetic damping without needing oscilloscopes or precision coils. The authors note the method is simple and claim decent results, so the educational intent comes through clearly. That part is straightforward and honest about what the apparatus can do with minimal gear. The real limitation sits in the formula itself. Terminal velocity for this geometry depends on pipe conductivity, magnet moment, radius, and length, with μ0 entering only through the B-field expression. If conductivity or magnet strength come from tables or separate nominal measurements rather than being determined in the same run, the solved μ0 just absorbs those uncertainties instead of measuring the constant independently. The abstract asserts accuracy without showing data, error bars, or a direct comparison to the accepted 4π × 10^{-7} value, so it is difficult to judge whether the smartphone readings actually deliver what is claimed. Sensor calibration drift and video frame-rate limits add further unaddressed scatter. This kind of note is mainly for physics instructors who want accessible induction experiments. A reader hunting for new lab ideas or low-cost ways to illustrate eddy currents might pick up a usable trick or two. It does not move the measurement of fundamental constants forward or introduce new analysis. I would not send it to peer review. The central extraction step needs explicit in-situ checks on the other parameters plus a proper uncertainty budget before it could stand even as an educational methods paper.

Referee Report

3 major / 2 minor

Summary. The paper claims to present a simple and novel smartphone-based method for determining the free-space permeability μ₀. The value is obtained from the terminal velocity of a magnet falling through a conducting pipe, detected using the phone's magnetic sensor and timed with a video player. The method is highlighted for its accuracy and precision.

Significance. Should the approach be experimentally validated with proper error analysis and independent parameter measurements, it would provide a valuable, accessible tool for physics education labs to measure a key electromagnetic constant using everyday technology. The integration of sensor data and video analysis demonstrates potential for reproducible student experiments.

major comments (3)

[Abstract] Abstract: The assertion of 'accuracy and precision of the results' is unsupported, as the manuscript supplies no experimental data, error bars, multiple trials, or direct comparison to the accepted value of μ₀ = 4π × 10^{-7} H/m.
[Method] Terminal velocity expression: Standard derivations give v_t ∝ mg / (σ B² r⁴) with B depending on the magnet moment and μ₀ (e.g., B = (μ₀/4π)(2m/r³)); the paper does not specify how conductivity σ, magnet moment, and geometry are measured independently in situ or how their uncertainties are propagated when solving for μ₀.
[Results] Experimental validation: No results section, table, or figure presents measured terminal velocities, calculated μ₀ values, sensor calibration details, or confirmation that the falling magnet has reached terminal regime, undermining the central claim.

minor comments (2)

[Abstract] The phrasing 'This method deserves its importance' in the abstract is awkward and should be revised for clarity.
[Title] Title and abstract should use the standard term 'permeability of free space' or 'magnetic permeability μ₀' for precision.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us identify areas where the manuscript can be strengthened. We have revised the paper to incorporate experimental data, clarify the methodology, and address all major points raised.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion of 'accuracy and precision of the results' is unsupported, as the manuscript supplies no experimental data, error bars, multiple trials, or direct comparison to the accepted value of μ₀ = 4π × 10^{-7} H/m.

Authors: We agree that the abstract's claim of accuracy and precision was not supported by data in the original submission. The revised manuscript includes a results section with multiple trials, measured terminal velocities with error bars, calculated μ₀ values, and a comparison to the accepted value. The abstract has been updated to reflect these additions without overstatement. revision: yes
Referee: [Method] Terminal velocity expression: Standard derivations give v_t ∝ mg / (σ B² r⁴) with B depending on the magnet moment and μ₀ (e.g., B = (μ₀/4π)(2m/r³)); the paper does not specify how conductivity σ, magnet moment, and geometry are measured independently in situ or how their uncertainties are propagated when solving for μ₀.

Authors: The referee correctly identifies the standard terminal velocity dependence. In the revised methods section, we now detail the independent measurements: conductivity σ of the pipe was measured separately using a four-point probe method, the magnet moment was calibrated in situ with the smartphone sensor against a reference field, and geometry parameters were measured with digital calipers. We have added the explicit error propagation analysis for μ₀ derived from these quantities. revision: yes
Referee: [Results] Experimental validation: No results section, table, or figure presents measured terminal velocities, calculated μ₀ values, sensor calibration details, or confirmation that the falling magnet has reached terminal regime, undermining the central claim.

Authors: We acknowledge that the original manuscript did not include a results section or supporting data. The revised version adds this section with tables of measured terminal velocities from repeated trials, figures showing the approach to terminal velocity, sensor calibration procedures, and the resulting μ₀ values with uncertainties. This provides the experimental validation for the method. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method relies on independent measurements and standard external formula

full rationale

The paper computes μ0 via the terminal-velocity expression for a magnet falling in a conducting pipe, with velocity and magnetic-field data obtained directly from smartphone sensor and video timing. This uses an established electromagnetic drag formula plus in-situ experimental inputs rather than any self-defined quantity, fitted parameter renamed as a prediction, or load-bearing self-citation. No reduction of the claimed result to its own inputs by construction is present in the described chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; central claim rests on the validity of the terminal-velocity expression and the assumption that smartphone measurements are sufficiently precise.

axioms (1)

domain assumption Terminal velocity of the falling magnet is governed by an expression that includes the permeability of free space as a parameter.
Directly referenced in the abstract as the source of the computed value.

pith-pipeline@v0.9.0 · 5336 in / 1029 out tokens · 33965 ms · 2026-05-08T18:36:32.225505+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Constants (RS derives c, ℏ, G as φ-powers; μ₀ is not on the RS ladder) reality_from_one_distinction (no overlap) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

k = (15/1024) μ₀² m² σ (1/a³ − 1/b³); kv_T = Mg

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 1 canonical work pages

[1]

Parameters a and b denote the inner and outer radii of the pipe, respectively

𝜇0 2𝑚2𝜎 ( 1 𝑎3 − 1 𝑏3), (2) Here, 𝜎 represents the conductivity of the material composing the pipe. Parameters a and b denote the inner and outer radii of the pipe, respectively. Additionally, μ0 represents the permeability of free space, while m stands for the magnetic moment of the magnet. Now, we can write 𝑘𝑣𝑇 = 𝑀𝑔. (3) 3 After a considerable time, tho...
[2]

Teach 82(9) 32-40

Vieyra R, Vieyra C, Jeanjacuot P, Marti A C 2015 Turn your smartphone into a science laboratory The Sci. Teach 82(9) 32-40

2015
[3]

Wright K 2020 Smartphone Physics on the Rise Physics 13 68

2020
[4]

Stampfer C, Heinke H, Staacks S 2020 A lab in the pocket. Nat. Rev. Mater. 5 169–170

2020
[5]

Pal S K, Sarkar S, and Panchadhyayee P 2024 LiDAR based determination of spring constant using smartphones The Physics Educator 2450001 doi: 10.1142/S266133952450001X Distance between two smartphones (m) Time interval between two peaks (s) Terminal velocity, 𝑣𝑇 (ms-1) 𝑘 = 𝑀𝑔 𝑣𝑇 (kg.s-1) μ0 (10-7 Hm-1) Average value of μ0 (10-7 Hm-1) 0.520 8.33 0.0624 ...

work page doi:10.1142/s266133952450001x 2024
[6]

Roy M K, Harbola M K and Verma H C 2007 Demonstration of Lenz’s law: Analysis of a magnet falling through a conducting pipe Am. J. Phys. 75 728–730

2007
[7]

Behroozi F 2018 Weighing a magnet as it falls with terminal velocity through an aluminium pipe Phys. Teach. 56 475-477

2018
[8]

Marín-Sepulveda C F, Castro-Palacio J C, Giménez M H, and Monsoriu J A 2023 Acoustic determination of g by tracking a freefalling body using a smartphone as a ‘sonar’ Phys. Educ. 58 035011

2023
[9]

Pal S K, Sarkar S, and P anchadhyayee P 2024 Determination of the magnetic moment of a magnet by letting it fall through a conducting pipe Phys. Educ. 59 015022

2024