3.1. An Old Faraday Idea

From 1891 to 1897, Pieter Zeeman carried out a series of experiments in Leyden under the supervision of both Kamerlingh Onnes and Lorentz. In particular, he was looking for an interference between the electromagnetic field and the frequency of light. Lorentz, in the same period, was after a unified theory of matter, light and electromagnetism. But Zeeman's interest referred to Faraday's old project based on the observation that a magnetic field modifies the way in which light is propagated and reflected and therefore it is very likely to modify its frequency. Faraday attempted unsuccessfully to detect such an interference. According to Zeeman, that failure depended on the low quality of Faraday's equipment, and in his notebooks he observes that “it might worth while to try the experiment again with the excellent auxiliaries of spectroscopy equipment of the present time” (Kox Reference Kox1997, 140; italics mine).

Notice that when he started his experiments, the available theories suggested that the interference was not detectable at all.

In the fall of 1896, the assumption of Zeeman turned out to be right. A widening in the spectral lines of the light of Sodium emitted in a magnetic field was detected. Lorentz predicted that the widening depended on a splitting of the spectral lines. The splitting was observed in a successive phase. The theoretical explanation was based on mechanical application of the Lorentz Force. Before getting into the details of experiments and related predictions, let us explore the general picture of the phenomena peculiar to Lorentz's theory.

3.2. The Dutch Synthesis: Charged Particles within Maxwell's Electromagnetism

Theoretical physics at the moment in which the new phenomenology appeared was concerned with an account of the relation between ether and matter. Maxwell's electromagnetic theory was dramatically successful but nevertheless incapable of accounting for phenomena like optical dispersion, the magneto-optical effects discovered by Kerr and Faraday, the high transparency of metal sheets to light, and electrolysis. All of this required a clear understanding of the microstructure of matter, electricity, magnetism and light, possibly a unifying theory. Maxwell's electromagnetism was silent on this issue (see Arabatzis Reference Arabatzis, Buchwald and Warwick2001, 176). Precisely these concerns inspired Lorentz's memoir of Reference Lorentz, Zeeman and Fokker1892, the contribution in which the Lorentz force was formulated. The framework was based on a dualist ontology designed by combining the ontological independence of the charge from the ether with the idea that any electromagnetic phenomenon is a form of disturbance in a medium.Footnote ² The resulting picture represented the common microphysical structure of matter, electromagnetism, and light in terms of the dynamic of charged particles perturbing the ether. Their electric charge aside, particles were classical rigid bodies. Maxwell's framework became the theory of ether and the electromagnetic processes had to obey to Maxwellian constrains. The charged particles were supposed to be the only kind of matter interacting with the ether, and since the interactions were essentially electric in nature, the medium itself was completely stationary. Further, the framework embedded a notion of local time and the first formulation of Lorentz's transformations.

The theory was presented as a combination of six hypotheses from which the fundamental laws were derived. It is worth exploring these hypotheses because they provide a description of the properties of ether and charged particles, thus giving a clear clue to the kind of picture Lorentz had in mind.

1. (i) “Charged particles have inertial mass and weight. The charged particles are in part mechanical bodies to which the laws of motion apply”Footnote ³ (McCormmach Reference McCormmach1970, 459).

2. (ii) “[The theory] identifies potential energy of an electromagnetic system with its electric energy, which is, in electromagnetic units,

   
   where f, g, h is the dielectric displacement at each point of the ether. The dielectric displacement satisfies
   
   outside the charged particle, and
   
   inside, where ρ is the density of electric charge” (McCormmach Reference McCormmach1970, 464).

From (i) and (ii) the potential energy of an electromagnetic field is defined as electric energy and this turns to be related to the dielectric displacement in the ether. The dielectric displacement must satisfy (3), i.e., a condition on the conservation of the density of charge at any point of the displacement. Notice that this suggests a deep interplay between the mechanical and electrical features of the theory, since the conservation of charge density is ‘localized’ in the particle. Broadly speaking, the charge carrier plays the fundamental role in the picture and the charge carrier is a mechanical entity. Here the ether, the location of the electric interaction, is completely characterized in terms of electric energy. This corresponds to attributing to the ether only electrical interactions with charged particles.

1. (iii) “[C]harged particles behave like rigid bodies; moreover, each point of a particle preserves the same value of ρ, whatever its motion” (McCormmach Reference McCormmach1970, 464).

2. (iv) Define as follows “the total electric current u, v, w as $u=\rho \xi +(\delta f/ \delta t) $, $v=\rho \eta +(\delta g/ \delta t) $, $w=\rho \zeta +(\delta h/ \delta t) $, where ζ, η, ξ is the velocity of a given point of a charged particle” (McCormmach Reference McCormmach1970, 464; the u, v, and w are the vectors of the total current).

Before we move on to the laws of the theory, let me emphasizes some peculiarities.

First of all, (i) and (iii) define the particle as a rigid body with mass and weight, equipped with an electric charge whose density has to remain constant during motion. These mechanical properties will turn out to be crucial in the analysis of the Zeeman effect.

Further, the above four hypotheses provide an immediate element of continuity with Maxwell's theory since they allow one to retrieve the notion of currents as incompressible fluids. The incompressibility of currents was obtained in that framework, treating the currents as the result of a strain in the ether surrounding the conductor (Darrigol Reference Darrigol1994, 285). In the new theory the currents are due to the motion of charged particles whose charge density is conserved during the motion, and thus are incompressible.

Let us come back to the hypotheses.

1. (v) The kinetic energy of the system is defined in terms of magnetic energy and it depends upon the total current.

2. (vi) “[T]he location of each point of the ether participating in the electromagnetic motions of the system is determined by the positions of all of the charged particles and by the values of f, g, h at all points in the ether” (McCormmach Reference McCormmach1970, 465).

Adding to these assumptions D'Alembert's Law, it is possible to derive the equations of motion in the ether. Let us come to the most important contribution of the theory, the equations of the Lorentz Force. In the following, V is the speed of light, f, g, and h describe the dielectric displacement at each point in the ether, ζ, η, and ξ represent the velocity of a given point of a charged particle, and α, β, and γ represent the magnetic force:

Lorentz derived these equations mechanically; they described the force acting on a particle with charge density ρ moving in the ether. In particular, the first integral on the right side represents the electrostatic force, whereas the second integral expresses the force acting on single particle moving through the ether. It is interesting to notice that Lorentz could derive Fresnel coefficients from his system of equations (McCormmach Reference McCormmach1970, 465–467).

The mechanical approach not only seems to provide a reasonable basis for the searched unification but also from the very beginning saves all the preceding relevant results. Let us now examine the kind of explanation performed by this framework in the case of Zeeman's discovery.

3.3. The Interference Detected

In the fall of 1896 Zeeman put a piece of asbestos soaked with common salt and exposed to the flame between the arms of a Rumkorf magnet. A Rowland grate provided the analysis of the line spectra. He described as follows what he was able to detect:

So, the first outcome was a widening in the lines of the light emitted by the sodium. It is important to observe that no splitting of the lines was found at this stage. Zeeman himself informed Lorentz of the outcome and officially reported the result to the faculty. Two days later, Lorentz communicated his account. His model resolved the broadening of the lines in a more complex pattern of splitting, and described accurately the structure of each line. In this sense, the explanation was a prediction of an unobserved phenomenon.

3.4. The Analysis of the Pattern of Interference

The model provided is based on the Lorentz Force but the notation below is due to the new vectorial formulation that Lorentz obtained for (5) by the end of 1895:

where H and E are respectively the magnetic and electric field and e and v are the charge and mass of the particle (Lorentz [Reference Lorentz, Zeeman and Fokker1895] 1935–1939).

Let z be the direction of the field H; then the equations of motion for a charged particle rotating around the centre of the atom, once a magnetic field is applied, are

The general solution of (8) is

with a and p constants and the frequency $\omega _{0}=2\pi (k/ m) $.

At this point is possible to derive two solutions for each of (6) and (7) by means of which two more values of frequency are obtained:

where ω₀ is the unaltered frequency, since z is both the direction of the light and the direction of the field H (Kox Reference Kox1997, 142).

From the mechanical point of view, the model treats the electron as an harmonic oscillator; this is what is expressed in the equation (6), (7), and (8), if we ignore the electromagnetic terms added in the extreme right side in (6) and (7). Once a magnetic field is applied, the particle experiences a Lorentz force and the period of its motion is altered according to the relations expressed by the electromagnetic terms, i.e., a rigid body moving of harmonic motion to which is applied a force. More generally the idea is that the electric ion is responsible for light emission. It has an oscillatory motion with arbitrary direction in space. This motion can be resolved in three oscillatory components of the same frequency: a linear oscillation and two circular oscillations, the two circular being one clockwise and the other anticlockwise. When a magnetic field is applied because of Lorentz's force acting on the ion, we have two different outcomes, both depending upon the direction of the detection.

1. (a) If the direction of detection is parallel to the direction of motion of the particles we have the pattern of splitting described above: two lines one broader than the other since the component in the direction of the field remains unaltered. The two lines observed together are wider than the line in absence of the field, i.e. precisely Zeeman's first observation.

2. (b) If the direction of detection is perpendicular to the direction of the motion of the particle a “triplet” is supposed to be detected. The pattern of interference has to involve the three component, with three plane polarized lines. The middle component has to have a direction parallel to the field and the other components have to maintain a direction perpendicular to the field.

This is exactly what Zeeman detected in a further experiment with the blue line of Cadmium, and this was not the only result (Kox Reference Kox1997).

3.5. The Ratio e/m of the Electron

Once the model had allowed one to calculate the alteration in the frequencies, the terms in the system of equation above were all corresponding to known values and therefore extracting the ratio e/m of the particle was just a matter of calculation:

If the change of the period is represented by δT, then [with $\delta (1/ T) =-\delta T/ T^{2}$] the positive or negative change of T is given by:

If δT is measured during the experiment, and H and T are known, the ratio e/m of the electron may be determined by the aid of formula (3) … . From the measured widening by means of the equation (3) the ratio e/m may now be deduced. It thus appears that e/m is of the order of magnitude 10⁷ [emu/g]. (Zeeman Reference Zeeman1913, 35)

Interestingly, the expectations were not towards a new constituent of matter but rather it was expected to find evidence in favor of the assumption that the charged particle was a Faraday's ion, which would lead to a broad unification of the field (Kox Reference Kox1997, 142).

Notice that the value of e/m is interpreted in strict relation with the notion of particle as rigid body since it is taken to provide an estimate of the size of the particle.