"Quantum" Forces and Electric Motor Design

G.R.Dixon, August 31, 2006

1. The Ampere/Lorentz Paradox.

Not long after Oersted discovered that a current-conducting wire can make the needle of a compass swing away from Earth’s north pole, Ampere found that two parallel, current-conducting wires attract or repel each other (depending upon whether the currents are in the same or in opposite directions). When Lorentz found that the magnetic force on a moving, charged particle is proportional to the cross product of the charge’s velocity and the external magnetic field that the charge moves through, the electromagnetic basis of the Amperian force became partially clear. For each wire in the Amperian pair theoretically has a magnetic field whose lines form concentric circles around the wire. Charge carriers in the other wire move
perpendicular to this magnetic field, and __v__ X __B__ points toward or away from the other wire.

All things considered, however, the Lorentz formula for magnetic forces does not alone explain Ampere’s discovery. For the current-conducting wire can feel a net force only if the individual, deflected charge carriers stay confined to the wire. Somewhat like the fact that atoms persist in time, something more than Maxwellian theory is needed in order to explain what Ampere observed (and what generations of electric motor designers have since exploited). In this article, this extra-Maxwellian player will be dubbed the "quantum force."

2. Historic Roots of the Quantum Force.

In the late 19’th century Hertz found that the passage of a spark, between metallic electrodes (cathode and anode) at different potentials, is facilitated if the cathode is irradiated with ultraviolet light. (In the present context the cathode will be the source of electrons.) Hallwache, Stoletov and Lenard subsequently found that charged particles are emitted (or ejected) from metallic surfaces similarly irradiated … a phenomenon dubbed the "photoelectric" effect. In 1900 Lenard, using techniques developed by Thomson to measure charge-to-mass ratios of charged particles, determined that the ejected particles are electrons.

The direction of ejected electrons opposes the direction of the incident light. Lenard (a) attracted the "photoelectrons" to an anode, and (b) turned them back toward the emitting metal using a cathode. Regarding the latter, he dubbed the magnitude of the negative potential, sufficient to turn __all__ of the photoelectrons back, the "stopping potential" (abbreviated V_{o}):

. (2_1)

Lenard also noted that, when ejected electrons are collected by an anode, then the size of the photoelectric current is proportional to the intensity of the incident light … a classically expected result. However, he also made the seminal discovery that __no__ electrons are ejected when the frequency of the incident radiation lies below a metal-specific threshold, __regardless of that radiation’s intensity__! And he found that when electrons __are__ ejected, their maximum speed (v_{max} in Eq. 2_1) __is proportional to the frequency of the incident radiation__, and not at all dependent on the incident radiation’s intensity! Indeed for frequencies greater than the threshold, ejection occurs even at very low light intensities. And ejection occurs more or less instantly, notwithstanding the fact that it would take minutes (or even hours) for the Poynting vector to integrate to m_{e}v_{max}^{2}/2 over the cross section of an atom.

In 1905 Einstein suggested a mechanism for these decidedly unclassical behaviors. Planck had previously concluded that atomic energies are quantized, and that atoms change energy levels more or less instantaneously when they absorb/emit radiation. Planck found the difference between adjacent atomic energy levels to equal hf, where f is the frequency of the absorbed/emitted radiation and h is a tiny constant (given his name). In considering the photoelectric effect, Einstein concluded that radiant energy must ultimately be __corpuscular__, at least in the sense that it appears to be absorbed/emitted instantaneously and in small but finite amounts. These light corpuscles (or quanta) were dubbed "photons."

In order to account for the fact that frequencies below the threshold frequency do not result in the complete escape of any photoelectrons, he proposed a metal-specific energy dubbed the "Work Function" (abbreviated "W"). Hypothetically it is only when the energy of incident photons exceeds the Work Function energy that electrons will have enough kinetic energy to completely break free from the irradiated surface. Assuming all of an incident photon’s energy is manifest as electron kinetic energy, the maximum kinetic energy of the ejected photoelectrons should equal the energy of the incident, ejection-causing photons __less__ the Work Function energy:

. (2_3)

In a series of experiments from 1914 to 1916 Millikan confirmed Eq. 2_3 (now known as Einstein’s Equation), finding that

. (2_4)

Einstein’s Equation (or more accurately his Work Function) is a remarkably simple and quasi-classical model for what must ultimately be a problem in quantum theory. For present purposes we shall assume that the Work Function idea is consistent with the fact that conduction electrons tend to remain confined to current-bearing wires, despite the fact that __Lorentz magnetic forces__ may deflect them and give them velocities perpendicular to the wire’s surface. Assuming the Work Function energy equates to a force acting through a distance, we shall refer to this Work Function force as a "quantum" force.

3. The Role of Quantum Forces in Ampere’s Discovery.

The direction of the Amperian deflecting force is consistent with the magnetic part of the Lorentz force:

. (3_1)

However, the rate at which such a force can do work is zero:

. (3_2)

Thus the magnetic force can do no work. Yet the wire can be linked to a mechanical load and, if free to move, can do mechanical work … a fact exploited by electric motors.

How can Eq. 3_2 be reconciled with the work done by electric motors? Answer: enter the quantum force. Fig. 3_1 depicts a drifting conduction electron that has been deflected by an external __B__ field. In the absence of any retaining quantum force, the electron might escape (at least for a small circular radius) from the wire.

Figure 3_1

Deflected Conduction Electron

Thanks to the quantum force, the electron experiences a decelerating force at the wire’s surface. Analogous to being caught by the end of a spring (whose other end is anchored to the wire’s atomic lattice), the electron is brought to a halt and is then accelerated back into the wire (being deflected back in the direction of the drift velocity by the same magnetic field). Twice the outward momentum is impulsively transferred to the wire, and collectively such impulses are manifest as the force first observed by Ampere.

Let us first consider the case where the wire is constrained from moving. Neglecting any radiation the electron might emit, and neglecting any lattice vibrations induced by the electron’s momentum reversal, the electron should return back into the wire with the same magnitude momentum as it attempted to leave with. No work is done on the __stationary__ atomic lattice. But the magnetic deflections, coupled with the theoretical momentum-reversing quantum forces, result in a net, constant lateral force on the wire.

If the wire/lattice moves with a velocity __V__ in the direction of the mean momentum that the magnetically deflected electron’s have when that momentum is reversed by the quantum forces, then in any velocity reversal –__F__ dot __V__ is __greater than zero__, and a net positive power is expended on the wire. Since the magnetic force can do no work, this power evidently comes from a decrease in electron kinetic energy each time
the direction of its momentum is reversed by the quantum force. Assuming the wire current does not decrease with time, each electron’s kinetic energy must be increased again following a momentum reversal. And these myriad, energy-increasing impulses can only be traced back to the driving emf (battery or whatever). Ultimately, then, it is the driving emf’s power expenditure that equates to the power delivered by the wire to a mechanical load (plus any Ohmic-generated heat energy).

Evidently the most efficient motor occurs in those cases where the quantum force engages, on average, when the magnetically deflected electron’s velocity is normal to the wire’s surface. In such cases the maximum amount of drift kinetic energy is converted to work done on the motor’s mechanical load. (In reality the decelerated electron probably induces some vibration in the wire’s atomic lattice, and some of the electron’s kinetic energy is also manifest as quasi-Ohmic heating.) Evidently a key design requirement is that the deflected electron’s velocity be as normal as possible to the wire’s surface when the quantum force engages. Since the electron’s trajectory generally has a small radius of curvature (owing to its tiny mass), utilization of very thin wire seems advisable. In cases where a Faraday disc is used in a homopolar motor, a "sunburst" of thin wires, radiating from a central hub to a peripheral collecting ring (and embedded in a dielectric disc for rigidity), might produce greater torques than a simple conducting disc.