STM-D-0505Paper2022Published and peer-reviewed

A Charged Particle Model based on Weber Electrodynamics for Electron Beam Trajectories in Coil and Solenoid Elements

Christof Baumgärtel · Simon Maher

Abstract and summary · read the original at the source

In one page

Wilhelm Weber wrote down a force law in 1846, fifteen years before Maxwell’s first paper on magnetic vortices: charges push and pull each other directly along the line joining them, with the strength depending on how fast their separation is changing. No field in between, no mediator. Christof Baumgärtel and Simon Maher at Liverpool test whether that older picture still works as engineering. They take a cathode-ray tube out of an oscilloscope, wrap a 290-turn coil around it, run one amp through it, and steer a 2000-volt electron beam through the coil at eight different entry angles, reading the landing spot off the screen five times each. Then they predict those landings three ways: with Weber’s direct force, with the standard magnetic-field calculation, and with a commercial charged-particle package. All three land on the same numbers, and all three match the measurements. They then map Weber’s force across a solenoid and find it reproduces the magnetic field map exactly.

Why it matters hereChapter ten is about what the potentials and the field really are, and this paper is the cleanest kind of evidence in that argument: a bench experiment in which a direct-action account with no field at all predicts what a field calculation predicts, to the same accuracy. It keeps the older description alive as a working engineering tool rather than a historical footnote.

What it claims

01Weber’s 1846 force law acts along the line joining two charges and depends on their separation, the rate of change of that separation and its second rate of change. It obeys Newton’s third law, conserves linear momentum, energy and angular momentum, reduces to Coulomb’s law for charges at rest, and yields Ampère’s force when applied to current elements.Sect. 2.1, Eq. (1) and the paragraph following Eq. (5)
Settled physics
02The model computes the deflection of the beam directly as a force between the coil’s charges and the beam’s charges, never constructing an electric or magnetic field as an intermediate step — which is what makes it a genuine test of the direct-action formulation rather than a re-derivation of the field result.Sect. 2, opening; Sect. 2.1
Published and peer-reviewed
03The experiment is deliberately ordinary and repeatable: a Hameg 203-6 oscilloscope cathode-ray tube with an external electron-gun cable, a single 290-turn coil of 159.5 mm radius and 170 mm length carrying one amp, and a beam accelerated through 2000 volts to about 2.65 times ten to the seventh metres per second, entered at eight angles set by 5 mm steps of electrostatic offset and read five times each from the screen graticule.Sect. 3, Fig. 2 and Eq. (38)
Published and peer-reviewed
04The three predictions agree with each other to hundredths of a millimetre across all eight entry angles — for the largest offset, 3.11 millimetres horizontal and 38.91 vertical from Weber, the same from the field model, 3.15 and 38.90 from the commercial package — and all three agree with the measured deflections. The direct-action model is equally as accurate as the field-based simulations.Sect. 4, results table and Fig. 3
Published and peer-reviewed
05Mapped across a solenoid cross-section, the total Weber force scaled by charge and electron velocity gives a quantity in tesla whose map is the same as the magnetic field magnitude, qualitatively and quantitatively; component by component the axial field matches the vertical force and the radial field matches the horizontal force, with the radial component carrying most of the influence in the fringing region.Sect. 5, Eqs. (39) and (40); Figs. 4, 5 and 6
Published and peer-reviewed
06The authors conclude that the two descriptions are indistinguishable in the near field at low velocity, and name where the direct-action version earns its keep and what comes next: complicated geometries where no field solution is convenient, low-energy decelerators and storage rings, and then space-charge effects added by superposition of particle-particle forces or by the virial theorem, plus tune and chromaticity shifts.Sect. 6, Conclusion
What to watch

Read it · abstract

Abstract

To aid with the design, evaluation, and optimisation of charged particle instrumentation, computer modelling is often used. It is therefore of interest to obtain accurate predictions for trajectories of charged species with the help of simulation. Particularly for solenoids and coils, which are often used for guiding, deflecting or focussing particle beams, knowledge of the magnetic field is required, especially in the fringing field regions. A novel model, which is based on a direct-line-of-action force between interacting charges, is described in this paper which accurately predicts the deflection of an electron beam trajectory traversing through a coil and the fringe field region. The model is further compared with a standard field model and a commercially available software package. Additionally, a relatively straightforward experiment has been designed and implemented to verify the simulation results, where it is found that the presented direct-action model is equally as accurate as field-based simulations compared with the experimental results. Furthermore, the magnetic field of a solenoid is visualised and analysed in terms of its radial, axial, and total field strength and compared to a force map obtained from the direct-interaction model. This representation allows for further comparison of the field and force interaction models and it is found that they are qualitatively the same.

Christof Baumgärtel and Simon Maher, School of Electrical Engineering, Electronics and Computer Science, University of Liverpool. Progress In Electromagnetics Research C, Vol. 123, 151–166 (2022).

(Abstract only. The complete article is free to read at the publisher — see the rights note for why the full text is not reproduced here, and for the two companion sheets in this library.)

The way in

https://doi.org/10.2528/pierc22061508Licence checked in both places. The published article, Progress In Electromagnetics Research C, Vol. 123, 151–166 (2022), carries no licence statement anywhere in its sixteen pages, and the journal’s article page at jpier.org carries only ‘Copyright © 2022 The Electromagnetics Academy. All Rights Reserved.’, checked on 2026-09-08 — no Creative Commons statement in either. So this sheet carries the summary, the claims and the authors’ own abstract and sends the reader to the source. The claims below were written from the complete published article, and the locators use its own section, equation and figure numbering. Received 15 June 2022. Baumgärtel and Maher write from the School of Electrical Engineering, Electronics and Computer Science at the University of Liverpool; Baumgärtel’s funding came from that school. Two companion sheets in this library carry the surrounding work, both reproduced in full under CC BY: the same authors’ review ‘Foundations of Electromagnetism: A Review of Wilhelm Weber’s Electrodynamic Force Law’, Foundations 2, 949–980 (2022), at /library/stm-bc8f1e5c8c; and Steffen Kühn’s ‘The importance of Weber-Maxwell electrodynamics in electrical engineering’ at /library/stm-717b3fa832.

How to cite it

Christof Baumgärtel, Simon Maher (2022) A Charged Particle Model based on Weber Electrodynamics for Electron Beam Trajectories in Coil and Solenoid Elements. doi:10.2528/pierc22061508

Where it sits in the curriculum

Scalar waves and the field behind the fields Inertia and gravity from the vacuum