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

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.