STM-D-0406Paper2015Published and peer-reviewed

Evaluation of Electron Beam Deflections across a Solenoid Using Weber-Ritz and Maxwell-Lorentz Electrodynamics

Ray T. Smith · Fred P. M. Jjunju · Simon Maher

Abstract and summary · read the original at the source

In one page

Send an electron beam past a long solenoid carrying a steady current and it bends. Textbooks work out that bend from the magnetic field where the beam is, through the Lorentz force. Ray Smith, Fred Jjunju and Simon Maher work it out a second way as well, using the force law Wilhelm Weber wrote in 1848 and Walther Ritz refined sixty years later: two charges act on each other directly along the line joining them, with corrections for their relative velocity and acceleration, and no field is constructed anywhere in the calculation. They measured real deflections on a standard Teltron beam tube for three double-wound solenoids of 0.25, 0.50 and 0.75 metres, each carrying 5.00 amps. For the two long solenoids both methods match the measurement. For the short one, where the field around the beam is not uniform, the field calculation predicts 6.4 millimetres against a measured 9.0, while the direct-action calculation gives 9.4 — because it sums forces over the actual turns of the coil instead of assuming a uniform field.

Why it matters hereChapter 10 is about what the potentials and the phase of a field really are, and this experiment sits right on that question: the authors point out that their calculation gives a finite classical force on the beam even for very long solenoids, where the standard account of the Aharonov-Bohm effect is written in terms of the vector potential and quantum mechanics. Chapter 3 gets the deeper stake, because a direct-action law that already carries the relative-velocity terms puts the usual relativistic mass correction in play as an interaction effect rather than a property of the particle. Read it beside the same group’s later fringing-field measurement at /library/stm-12fc74b8c7, the charged-particle model built on the same force law at /library/stm-b69dd44016, and the review of Weber’s law at /library/stm-bc8f1e5c8c.

What it claims

01Where the field around the beam is not uniform, the two calculations part company and the direct-action one is right. For the shortest solenoid, 0.25 m long, the measured mean deflection was 9.0 plus or minus 1.0 mm; the Weber-Ritz calculation gave 9.4 plus or minus 0.5 mm and the Lorentz-force calculation 6.4 plus or minus 0.5 mm.Table 1 and Sect. 5, Results
Published and peer-reviewed
02Where the field is uniform the two agree, which is what makes the short-solenoid gap meaningful rather than an artefact. For the 0.50 m solenoid the measurement was 3.8 plus or minus 1.0 mm against 3.9 from Weber-Ritz and 3.2 from Lorentz; for the 0.75 m solenoid it was 1.5 plus or minus 1.0 mm against 1.6 and 1.5. Hall-probe mapping and the calculation of Farley and Price both confirm the external field of the two longer solenoids is uniform along the beam to within about 10 per cent, and that the shortest one is not.Table 1, Fig. 6, and Sects. 3.3.1 and 5
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03The Weber force acts along the line joining two charges and depends on their separation, their relative velocity and their relative acceleration, reducing to Coulomb’s law for charges at rest. Summing it over a current loop, the electron’s attraction to the fixed positive ions cancels the electrostatic part and leaves a purely relative-motion term proportional to the product of the beam and drift velocities divided by the square of the speed of light; the deflection then follows by equating the impulse of that non-uniform force to the electron’s vertical momentum change. Newton’s third law is inherently obeyed by Weber’s law, so momentum and angular momentum are conserved.Sects. 2, 3.1 and 3.3.2, Eqs. 1 to 6; Concluding remarks
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04The apparatus is deliberately ordinary and the controls are stated. A Teltron electron beam tube at 1200 V gives electrons about 2.06 times 10 to the 7th metres per second, crossing the 10 cm deflection zone in 4.85 nanoseconds; the three solenoids are doubly wound at 2600 turns per metre, about 6 cm in diameter, each carrying 5.00 A. The tube was aligned north-south to minimise the Earth’s field, the solenoid current was reversed and the mean taken to remove a residual deflection of about 1 mm, and the whole scale was calibrated against Helmholtz coils at close to 4.2 microtesla per milliamp.Sect. 4, Experimental investigation, and Fig. 5
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05A piece of folklore is corrected along the way. It is often repeated that an infinite solenoid or a toroid carrying direct current has no external magnetic field, and that is false: standard theory gives an azimuthal field outside an infinite solenoid falling as the inverse of the distance, and finite solenoids leak a measurable axial field. Measured with a Hall probe at 5.00 A, that field is about 99 microtesla along the beam for the 0.50 m solenoid and about 44 for the 0.75 m one — and the beam is sensitive enough that the Earth’s field alone, around 20 microtesla, moves the spot by about 2 mm as the free-standing tube is rotated.Sect. 4, opening paragraphs, and Fig. 6
Settled physics
06The authors call this a preliminary investigation and name the next measurements. They ask for larger pulses of direct current to shrink the error bars and for much smaller diameter solenoids to probe the Aharonov-Bohm question, where Boyer argues for a semi-classical account of the phase shift and Caprez, Barwick and Batelaan’s time-of-flight test found no delay. They also propose a purpose-built tube reaching electron speeds near light, which would test Ritz’s theory in the regime where the Lorentz calculation needs its relativistic correction: on O’Rahilly’s reading, Ritz matches the high-speed data with the constant lambda set to 3 rather than minus 1, and without applying that correction directly.Sect. 6, Concluding remarks, final three paragraphs
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Read it · abstract

Abstract

The deflection of charged particle beams by electric and/or magnetic fields is invariably based on the field centred approach associated with Maxwell-Lorentz and incorporated into the Lorentz force formula. Here we present an alternative method of calculation based on the force formula of Weber-Ritz and which does not involve, directly, the field entities E and B. In this study we evaluate the deflection of an electron beam by a long solenoid carrying direct current and positioned centrally across the beam. The experiment has some bearing on the Aharonov-Bohm effect in that our calculations indicate that even for very long solenoids the classical force on the beam remains finite. The standard interpretation of the effect is, however, in terms of quantum mechanics and vector potential. Experimental measurements have been made of electron beam deflections by three solenoids, 0.25 m, 0.50 m and 0.75 m long; each solenoid is doubly wound with the same winding density (2600 turns per metre) and carrying the same current of 5.00 A d.c.. Our results indicate that, within the limits of experimental error, both Weber-Ritz and Maxwell-Lorentz theories correlate with measurements for the longer solenoids. However in the case of the shortest solenoid, the lack of uniformity of the magnetic field, leads to significant error in the calculation of beam deflection by the Lorentz force. By contrast in a Weber-Ritz calculation a precise value of beam deflection is obtained by equating the impulse of the non uniform beam force to the vertical momentum change of the electron. This is a fundamentally different approach which uses a statistical summation of forces on the beam in terms of relative velocities between moving electrons and involves a direct computation of the vertical force on the beam due to the circling solenoid current. This method has distinct advantages in terms of economy; that is, it does not involve directly field entities E and B, nor the leakage flux from the solenoid or the vector potential.

Ray T. Smith, Wirral Metropolitan College; Fred P. M. Jjunju and Simon Maher, Department of Electrical Engineering and Electronics, University of Liverpool. Progress In Electromagnetics Research 151, 83–93 (2015).

(Abstract only. The complete paper is free to read at https://www.jpier.org/PIER/pier.php?paper=15021106 and via https://doi.org/10.2528/pier15021106 — see the rights note for why the full text is not reproduced here.)

The way in

https://doi.org/10.2528/pier15021106Published as Progress In Electromagnetics Research 151, 83–93 (2015); received 11 February 2015, accepted 3 April 2015, scheduled 23 April 2015. Ray T. Smith is at Wirral Metropolitan College; Fred P. M. Jjunju and Simon Maher, the corresponding author, are in the Department of Electrical Engineering and Electronics at the University of Liverpool. TITLE. The skeleton for this page carried the title in the publisher’s all-capitals house style; it is set here in the paper’s own running-head case. LICENCE. The journal is free to read and the article PDF downloads without a paywall, which is why aggregators record it as gold open access, but no Creative Commons statement appears in the article or on the journal site — both the PDF and the publisher’s pages carry only ‘Copyright, The Electromagnetics Academy, All Rights Reserved’ — so the page stays abstract-only. TEXT. The abstract below is the article’s own, taken from the publisher PDF; the extraction dropped the fi, fl and ff ligatures and they are restored. The summary and the claims were written from the complete eleven-page paper, and the section, table, figure and equation numbers in the locators are the paper’s own. Equations are described in words because the page is MDX.

How to cite it

Ray T. Smith, Fred P. M. Jjunju, Simon Maher (2015) Evaluation of Electron Beam Deflections across a Solenoid Using Weber-Ritz and Maxwell-Lorentz Electrodynamics. doi:10.2528/pier15021106

Where it sits in the curriculum

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