Not All Uniform B-Fields Are The Same!

Benjamin K. Luna; Murray S. Daw

Not All Uniform B-Fields Are The Same!

T0 review · 0 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read When a uniform magnetic field is ramped off, a charge's final momentum is set by the vector potential at its location, so identical B-fields from different sources produce different dynamics.

desk verdict Careful worked examples showing uniform B-fields from different sources give different impulses and torques; the main insight is not new, but the explicit calculations are, and the only real flaw is in the point-charge section. read the letter →

arxiv 2508.01518 v1 pith:BPFOMT2K submitted 2025-08-02 physics.class-ph

Benjamin K. Luna , Murray S. Daw This is my paper · ORCID

classification physics.class-ph

keywords vectorpotentialelectromagneticinductionuniformmagneticfieldquasistaticapproximationforcelawinformhiddenmomentumchargedcapacitorcausalstructureofelectrodynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a standard induction problem—a charged capacitor in a uniform magnetic field that is then ramped down to zero—is not well posed unless the source of the field is specified. The reason is that the final momentum of a stationary charged body is set by the initial vector potential at its location, $\mathbf{p}_f = (q/c)\mathbf{A}(\mathbf{r},0)$, not by the local magnetic field. Different source configurations that produce the same uniform $\mathbf{B}$ over the capacitor region therefore produce different total impulses and different torques on the plates, as shown for planar-current and solenoid sources. This matters because it identifies the vector potential as a physical carrier of source information and reframes a famous textbook exercise in terms of causal structure rather than field bookkeeping.

What carries the argument

The load-bearing object is the quasistatic vector potential $\mathbf{A}(\mathbf{r},t) = (1/c)\int \mathbf{J}(\mathbf{r}',t)/|\mathbf{r}-\mathbf{r}'|\,d^3r'$, together with the force law in potential form, $\mathbf{F} = q(-\nabla\phi - (1/c)\partial\mathbf{A}/\partial t + \mathbf{v}\times(\nabla\times\mathbf{A}))$. In the high-mass, quasistatic limit the velocity term is negligible and $\phi$ is static, so the entire force is the induction field $\mathbf{E}_{\rm ind} = -(1/c)\partial\mathbf{A}/\partial t$; integrating over the ramp gives $\mathbf{p}_f = (q/c)\mathbf{A}_i$. The machinery converts the causal chain—currents generate $\mathbf{A}$, then changing $\mathbf{A}$ pushes charges—into a direct formula for impulse and torque that keeps the source explicit.

What would settle it

Build the same uniform $\mathbf{B}$ field with two different sources—opposing planar currents and a solenoid—place identical charged capacitors inside, ramp both down identically, and compare impulses and plate angular momenta. The paper predicts the solenoid case has half the total impulse of the planar-current case and different per-plate torques; if the measured impulse and torque distribution are the same for both sources, the central claim that the vector potential carries the dynamical information would be falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that in quasistatic induction the dynamical effect of switching off a magnetic field is carried entirely by the vector potential: for a charged body initially at rest and heavy enough not to move appreciably, the final momentum is $\mathbf{p}_f = (q/c)\mathbf{A}_i$, where $\mathbf{A}_i$ is the initial vector potential at the body. Because $\mathbf{A}$ is an integral over the current source, it contains information about where and how the currents are arranged, while the local field $\mathbf{B} = \nabla\times\mathbf{A}$ does not. The paper demonstrates this by placing the same charged capacitor in the same uniform $\mathbf{B}$ field produced by different sources: opposing planar currents (centered or shifted) give the same total impulse but different angular momentum; the same currents rotated by $90^\circ$ give zero net impulse but opposite plate torques; and a solenoid gives half the impulse of the planar-current case. It concludes that the common flux-form induction-law solution smuggles in a source assumption, and that the problem as originally stated is underdetermined.

Load-bearing premise

The key assumption is that the charged body hardly moves while the field is being turned off: the derivation keeps only the $-(q/c)\partial\mathbf{A}/\partial t$ force and drops the magnetic $q\mathbf{v}\times\mathbf{B}$ force, so if the capacitor accelerates appreciably during the ramp, Eq. (7) no longer describes its final momentum.

Editorial extensions

If this is right

The textbook exercise as normally stated is underdetermined: the answer changes with the source, so a well-posed version must specify how the uniform field is produced.
In identical uniform $\mathbf{B}$-fields, measured impulses and angular momenta can differ by factors of two or by even/odd distribution between plates, so integrating field momentum alone does not predict the motion of an object inside the field.
Teaching the vector potential as potential momentum per unit charge gives students a direct way to solve induction problems and explains why local field values are not causally sufficient.
The point-charge version of the problem, which has no convenient closed loop, is solved immediately by $\mathbf{p}_f = (q/c)\mathbf{A}_i$, showing that the approach generalizes beyond capacitor geometries.
A field-momentum conservation calculation, although it can account for the impulse, does not reveal which part of the source drives the motion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because $\mathbf{p}_f=(q/c)\mathbf{A}_i$ ties a measurable impulse to the vector potential at one point, a controlled field-ramp experiment could serve as a local probe of $\mathbf{A}$, a use the paper does not explicitly propose.
The same source-dependence critique applies to any quasistatic 'uniform field' problem: wherever only local fields are matched but source geometry is not, the dynamics is not unique, so the lesson generalizes beyond magnetic fields.
For a light test charge, Eq. (7) should be the leading term in an expansion; adding the $\mathbf{v}\times\mathbf{B}$ correction gives a solvable equation for a chosen ramp and a quantitative, testable extension that the paper leaves implicit.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The paper does what it says: it walks through the Griffiths 8.6 capacitor problem, shows the standard flux-form solution smuggles in an assumption about the source, and then computes the impulse and torque for three concrete source configurations (opposing planar currents centered and shifted, rotated planar currents, infinite solenoid) that all produce the same uniform B over the capacitor. The calculations are straightforward and internally consistent, and the torque results for shifted plates and for rotated sources are genuinely not in the earlier treatments I know. If someone is going to teach this problem, these examples will be useful.

The central pedagogical claim—that knowledge of the source is required to answer 'what happens' because the local B is not enough—is not new. It is the point of Babson et al., of Hu's arXiv paper, and of McDonald's note, and the paper frankly says so. What is new is the systematic set of explicit A-based computations and the torque values, and those are worth having.

The soft spot I'd want fixed before publication is Section IV. Eq. (7), p_f = q A_i/c, is derived by discarding the v×(∇×A) term on the grounds that the charge is massive and barely moves during the ramp. That is a legitimate limit, but it is not a universal relation for a point charge of arbitrary mass. During the ramp the accumulated magnetic deflection can be order-one even if the velocity stays small, and then the v×B force contributes to the impulse. Section IV applies Eq. (7) to a bare point charge without reimposing the high-mass condition, so as written it overstates the generality. The capacitor examples are largely insulated from this because the net v×B force on a rigid neutral capacitor cancels for translation, so the main claim of the paper still stands.

There are also minor limitations the authors already acknowledge: infinite idealized sources, quasistatic turnoff, fixed insulator plates with no mobile charges. Those are fine for a pedagogical paper.

Bottom line: a solid, honest contribution to the pedagogy literature, modestly novel in its worked examples. It deserves a serious referee, with attention to Section IV. I'd bring it to a reading group and would cite it if I were writing about this problem.

Referee Report

0 major / 3 minor

Summary. The manuscript revisits a textbook induction problem in which a charged parallel-plate capacitor sits in a uniform magnetic field that is turned off quasi-statically. The authors argue that the mechanical effect (impulse and torque) on the capacitor is not fixed by the local B-field alone but depends on the current-source configuration. Using the Lorentz force in potential form, F = q(-∇φ - (1/c)∂A/∂t + v×(∇×A)), and assuming the high-mass quasistatic limit so the v×(∇×A) term is negligible, they obtain p_f = qA_i/c (Eq. 7). They then compute the impulse and torque for three sources that produce the same uniform B-field at the capacitor: infinite planar currents (centered and shifted), the planar currents rotated by 90°, and an infinite solenoid. The results differ: the net impulse is -qdB0/c for the two planar-current arrangements, but -qdB0/(2c) for the solenoid, with different angular-momentum transfers in each case. Section IV extends the argument to a point charge, and the paper concludes that the source must be specified for the problem to be well-posed.

Significance. The paper provides a clean, analytically explicit demonstration that local field values are insufficient to determine induction forces; the source configuration matters. The derivations in Sections III.B-III.E are internally consistent: integrating -q/c ∂A/∂t over the ramp gives the stated impulses, and the torque expressions follow from the force distributions. The paper is transparent about its main idealizations (infinite sources, quasistatic ramp, high mass) and computes all results from a single, non-circular argument with no fitted parameters. If the caveats are stated precisely, the paper is a useful pedagogical contribution that makes a subtle and frequently missed point accessible.

minor comments (3)

[Section IV, Eq. (28)] The point-charge example applies Eq. (7) without repeating the heavy-mass/quasistatic restriction stated in the derivation of Eq. (7) (text before Eq. (6) and footnote [4]). For a finite-mass charge, the v×(∇×A) term in Eq. (5) becomes non-negligible once the charge is set in motion, so Eq. (28) is not the general final momentum; please add an explicit 'in the infinite-mass limit' qualifier or provide the finite-mass analysis.
[Section II.A and Section III.B] The label 'false' for Eq. (4) is misleading because Eq. (16), the correct result for the planar-current source, is numerically identical to Eq. (4). The flaw is the hidden assumption about the source in step (2), not the numerical value; suggest rewording 'false' to 'not generally valid' and noting that the value coincides with the planar-current answer.
[Section III.E] The infinite-solenoid example should explicitly state that the capacitor is placed well inside the solenoid and that the same quasistatic and high-mass assumptions used in Sections III.B-III.D carry over; currently this must be inferred by the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivation is self-contained and parameter-free; the heavy-mass caveat affects validity, not circularity.

full rationale

No load-bearing step reduces to its own inputs. Eq. (7) follows by integrating Eq. (6), which is the -q/c dA/dt term of the standard Lorentz force (Eq. (5)) under the explicitly stated quasistatic/high-mass approximation; the v x B term is dropped on a stated physical assumption (text before Eq. (6) and footnote [4]), not by assuming the conclusion. The different impulses and torques in Secs. III.B-III.E are computed from distinct source configurations using the independent retarded-potential formula (Eq. (12)), and the fact that different sources give different dynamics is the paper's claim, not an input. The authors explicitly warn that the source variations are not gauge transformations (footnote [27]), so the alternative A-fields encode genuinely different current sources rather than relabelings. There are no fitted parameters, no prediction reverse-engineered from a fitted subset, and no load-bearing self-citations: the cited supporting works ([2], [15]-[17]) are standard external references and do not supply a uniqueness theorem that forces the result. The finite-mass limitation of Eq. (7) raised by a skeptical reader is a scope condition on the high-mass limit, not circular reasoning.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; all quantities (B0, q, a, d, f) are given inputs. The derivation leans on standard retarded potentials and the Lorentz force, with domain assumptions about quasistatic turnoff and infinite sources.

assumptions (5)

domain assumption Quasistatic approximation: retarded potentials are replaced by instantaneous integrals (Eq. 12).
The turnoff is described as quasistatic; radiation and retardation are neglected.
domain assumption Infinite planar current sheets and infinite solenoid are used as sources.
These idealizations produce exactly uniform B; footnote [26] notes zeroth-order approximation for finite sheets.
domain assumption Charges are fixed on insulating plates of high mass.
Footnote [4] and Section III: plates do not move appreciably during the ramp, so v×B is negligible.
standard math Lorenz gauge retarded potentials (Eq. 11) are valid solutions.
Standard inhomogeneous wave equation solutions under the Lorenz gauge.
standard math Lorentz force in potential form, Eq. (5).
Standard classical electrodynamics force law.

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Cite this review

Pith. "Pith review of Not All Uniform B-Fields Are The Same!." pith.science (2026). https://pith.science/paper/BPFOMT2K

@misc{pith2026250801518,
  author       = {Pith},
  title        = {Pith review of: Not All Uniform B-Fields Are The Same!},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPFOMT2K}},
  note         = {Machine review of arXiv:2508.01518}
}

abstract

Set up a charged rectangular-plate capacitor in a uniform $\mathbf{B}$-field. Quasi-statically turn off the $\mathbf{B}$-field - what happens? We show that this seemingly simple induction problem (which appears in a famous undergraduate electrodynamics textbook) highlights the causal structure of electrodynamics. It may be (as it is in this problem) that different source configurations can produce the same uniform $\mathbf{B}$-field over some finite region, but very different dynamics when the source is turned off. We see also that the vector potential gives more information about the source of the $\mathbf{B}$-field, and is a very useful (and under-appreciated) way to understand induction and other key aspects of electrodynamics.

Figures

Figures reproduced from arXiv: 2508.01518 by the authors.

**Figure 2.** FIG. 2. A rectangular Faraday closed path with two sides of [PITH_FULL_IMAGE:figures/full_fig_p001_2.png] view at source ↗

**Figure 3.** FIG. 3. Infinite opposing planar currents, with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 5.** Figure 5: FIG. 5. Infinite opposing planar currents shifted down by a [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 4.** Figure 4: FIG. 4. The arrows shown are the net forces on the capacitor [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 6.** Figure 6: FIG. 6. The arrows shown are the net forces on the capacitor [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 7.** Figure 7: FIG. 7. Infinite opposing planar currents, but rotated [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 9.** Figure 9: FIG. 9. A solenoid of infinite length with [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

**Figure 10.** Figure 10: FIG. 10. The arrows shown are the forces on the parts of the [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

**Figure 11.** Figure 11: FIG. 11. The arrows shown are the initial uniform [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

Wu and Chen Ning Yang, Evolution of the Con- cept of the Vector Potential in the Description of Fun- damental Interactions , International Journal of Modern Physics A, v

A.C.T. Wu and Chen Ning Yang, Evolution of the Con- cept of the Vector Potential in the Description of Fun- damental Interactions , International Journal of Modern Physics A, v. 21, no. 16, pp. 3235-3277, 2006

work page 2006
[2]

Anthony Rizzi, Physics for Realists: Electricity and Mag- netism, IAP Press, 2011

work page 2011
[3]

John David Jackson, Classical Electrodynamics , 1962, John Wiley & Sons, Inc

work page 1962
[4]

Though the textbook source [6] for this problem does state it for a ’capacitor’, we have found it cleaner to demonstrate the principles involved using two uniformly charged insulator plates. The case of an insulator does not bring up the possible issue of mobile charges moving inside the plates, which would needlessly complicate the problem and make it mo...

work page
[5]

Specif- ically, one would like to know over what region the uni- form B-field is being specified

This vagueness should be disturbing to the reader. Specif- ically, one would like to know over what region the uni- form B-field is being specified. To have a unique answer to the question, we must specify some finite region within which the B-field is arbitrarily uniform. We delay this discussion for reasons that will become apparent shortly

work page
[6]

almost entirely wrong

(3rd edition only) in the chapter dealing with (among other things) the Poynting vector. [7] In particular, the line of solution suggested in that textbook was to calcu- late (approximately) the volume integral of the Poynting Vector between the plates and to show that the impulse imparted to the plates while the B-field is turned off can be calculated us...

work page
[7]

Griffiths, Introduction to Electrodynamics; 3rd ed., Prentice-Hall, 1999

David J. Griffiths, Introduction to Electrodynamics; 3rd ed., Prentice-Hall, 1999

work page 1999
[8]

The issues raised in this pa- per regarding the exercise above are not extraordinary - they are typical of the understanding of physicists and physics students

It should be noted that this textbook is excellent in train- ing students in the equations of electromagnetism, which is the reason for its just fame - a problem from this par- ticular book is only singled out to illustrate the generic point about all physicists’ understanding of electromag- netism broadly and would apply even more so to other authors of ...

work page

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[9]

McDonald, Electromagnetic Momentum of a Ca- pacitor in a Uniform Magnetic Field , 2023

Kirk T. McDonald, Electromagnetic Momentum of a Ca- pacitor in a Uniform Magnetic Field , 2023

2023
[10]

Ben Yu-Kuang Hu, On electromagnetic momentum of an electric dipole in a magnetic field , arXiv, 10.48550/arXiv.1408.4144, 2014

2014 doi arXiv
[11]

, Hidden momentum, field momen- tum, and electromagnetic impulse , American Journal of Physics, v

David Babson et al. , Hidden momentum, field momen- tum, and electromagnetic impulse , American Journal of Physics, v. 77, no. 9, pp. 826-833, 2009

2009
[12]

It was clarified by Hu in the above manuscript that find- ing the hidden momentum of the source of the B-field is not necessary to obtain the impulse on the capacitor
[13]

As the electron was not discovered until 1897, the understanding of a fundamental unit of charge at that time was not quite the same as the modern under- standing

In his treatise A Dynamical Theory of the Electromag- netic Field , Maxwell writes this equation in the form of an electromotive force, which would be force per unit charge. As the electron was not discovered until 1897, the understanding of a fundamental unit of charge at tha...
[14]

Niven, The Scientific Papers of James Clerk Maxwell, James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field , v

W.D. Niven, The Scientific Papers of James Clerk Maxwell, James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field , v. 1, pp. 559, Cambridge Li- brary Collection - Physical Sciences, Cambridge Univer- sity Press, 2011

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[15]

E. J. Konopinski, What the electromagnetic vector poten- tial describes, American Journal of Physics, v. 46, no. 5, pp. 499-502, 1978

1978
[16]

Semon and John R

Mark D. Semon and John R. Taylor, Thoughts on the magnetic vector potential , American Journal of Physics, v. 64, no. 11, pp. 1361-1369, 1996

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[17]

Bertrand Berche, Daniel Malterre, and Ernesto Med- ina, Gauge transformations and conserved quantities in classical and quantum mechanics , American Journal of Physics, v. 84, no. 8, pp. 616-625, 2016

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[18]

Konopinski, Electromagnetic Fields and Rela- tivistic Particles , 0-07-035264-X, International series in pure and applied physics , 1981, McGraw-Hill, Inc

Emil J. Konopinski, Electromagnetic Fields and Rela- tivistic Particles , 0-07-035264-X, International series in pure and applied physics , 1981, McGraw-Hill, Inc

1981
[19]

Andrew Zangwill, Modern Electrodynamics, 2013, Cam- bridge University Press

2013
[20]

The fourth edition very briefly mentions interpreting the vector potential as potential momentum
[21]

Griffiths, Introduction to Electrodynamics; 4th ed., 0-321-85656-2, 2013, Pearson Education, Inc

David J. Griffiths, Introduction to Electrodynamics; 4th ed., 0-321-85656-2, 2013, Pearson Education, Inc

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[22]

Jefimenko, Electricity and Magnetism: An In- troduction to the Theory of Electric and Magnetic Fields , Appleton-Century-Crofts, pp

Oleg D. Jefimenko, Electricity and Magnetism: An In- troduction to the Theory of Electric and Magnetic Fields , Appleton-Century-Crofts, pp. 516, 1966

1966
[23]

This is misspelled in Jackson [3] and various other treat- ments: the force law is named after Hendrik Lorentz (1853-1928), whereas the gauge condition is named af- ter Ludvig Lorenz (1829-1891)

1928
[24]

These solutions of the inhomogeneous wave equations re- quire that all of the sources are localized within some finite volume (that is, there are no sources ρ or J at in- finity)
[25]

(10), reduces to ∇ · A = 0, which is, interestingly, identical in form with the Coulomb gauge condition

It should be noted that since the scalar potential in the cases that we will consider does not change, the Lorenz gauge condition, Eq. (10), reduces to ∇ · A = 0, which is, interestingly, identical in form with the Coulomb gauge condition
[26]

Note that this is a zeroth-order approximation to a set of finite sheet currents in 1 /L, where L is half the length of each of the sides of the plates
[27]

Gauge differences do not and cannot imply differences in dynamics

Do not be deceived - this and the other changes in source mentioned do NOT constitute a difference of gauge. Gauge differences do not and cannot imply differences in dynamics
[28]

Even if one does not wish to use the vector potential, one cannot avoid the (at least implicit) consideration of the source, as one clearly sees in the expression for E in the Jefimenko equations
[29]

(11) (and Eq

In general, A will point in the direction of the cur- rent source (superposition), as is stated on page 248 of the fourth edition of Griffiths [21], and as can easily be sketched based on Eq. (11) (and Eq. (12) in the qua- sistatic condition)

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Wu and Chen Ning Yang, Evolution of the Con- cept of the Vector Potential in the Description of Fun- damental Interactions , International Journal of Modern Physics A, v

A.C.T. Wu and Chen Ning Yang, Evolution of the Con- cept of the Vector Potential in the Description of Fun- damental Interactions , International Journal of Modern Physics A, v. 21, no. 16, pp. 3235-3277, 2006

work page 2006

[2] [2]

Anthony Rizzi, Physics for Realists: Electricity and Mag- netism, IAP Press, 2011

work page 2011

[3] [3]

John David Jackson, Classical Electrodynamics , 1962, John Wiley & Sons, Inc

work page 1962

[4] [4]

Though the textbook source [6] for this problem does state it for a ’capacitor’, we have found it cleaner to demonstrate the principles involved using two uniformly charged insulator plates. The case of an insulator does not bring up the possible issue of mobile charges moving inside the plates, which would needlessly complicate the problem and make it mo...

work page

[5] [5]

Specif- ically, one would like to know over what region the uni- form B-field is being specified

This vagueness should be disturbing to the reader. Specif- ically, one would like to know over what region the uni- form B-field is being specified. To have a unique answer to the question, we must specify some finite region within which the B-field is arbitrarily uniform. We delay this discussion for reasons that will become apparent shortly

work page

[6] [6]

almost entirely wrong

(3rd edition only) in the chapter dealing with (among other things) the Poynting vector. [7] In particular, the line of solution suggested in that textbook was to calcu- late (approximately) the volume integral of the Poynting Vector between the plates and to show that the impulse imparted to the plates while the B-field is turned off can be calculated us...

work page

[7] [7]

Griffiths, Introduction to Electrodynamics; 3rd ed., Prentice-Hall, 1999

David J. Griffiths, Introduction to Electrodynamics; 3rd ed., Prentice-Hall, 1999

work page 1999

[8] [8]

The issues raised in this pa- per regarding the exercise above are not extraordinary - they are typical of the understanding of physicists and physics students

It should be noted that this textbook is excellent in train- ing students in the equations of electromagnetism, which is the reason for its just fame - a problem from this par- ticular book is only singled out to illustrate the generic point about all physicists’ understanding of electromag- netism broadly and would apply even more so to other authors of ...

work page