Historical research guide · 0505 · 2022
A Charged Particle Model based on Weber Electrodynamics for Electron Beam Trajectories in Coil and Solenoid Elements
Christof Baumgärtel · Simon Maher
In one minute
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.
Connection
Chapter 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.
From the source
Weber’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)
The 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
The 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)
The 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
Mapped 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
The 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
The way in
https://doi.org/10.2528/pierc22061508How 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