Magnetic Effects In Almost All Fundamental Experiments
Magnetic phenomena plays critical role in nearly all fundamental experiments in physics. Briefly we have:
- Oersted’s 1820 deflection of compass needle by a current.
- Faraday’s introduction of “electromagnetic” lines of force (1851) to explain his experiments on voltaic and magneto induction (1830s).
- Faraday’s electromagnetic lines of force are formalized by \(E,B\) in Maxwell’s equations (~1860s).
- Hall’s 1879 experiment deciding the sign of charge carriers in conductors.
- Zeeman’s 1897 experiment on the splitting of spectral lines in magnetic fields.
- Kaufmann’s 1901 experiment on deflection of electric particles in magnetic fields.
- Stern-Gerlach 1922 experiments on interaction of “spin” with magnetic moments.
However we insist that magnetism remains poorly understood. Even today efforts are made to “discover” magnetic monopoles, and it’s forgotten that Ampere (Assis and Chaib 2015, 19.2) demonstrated the equivalence of magnetic and electric forces, and Ampere [ampere, 19.1] demonstrated the nonexistence of magnetic monopoles via the elementary observation that \(div(curl)=0\). Thus Ampere found the idea of magnetic poles and dipoles as superfluous and disposable hypotheses.
Another fundamental critique of magnetic interactions due to Ampere (Assis and Chaib 2015, 19.3) is that these interactions are neither fundamental nor primitive. Indeed Oersted, Faraday, Maxwell, et. al., all postulate interactions between current carrying wires and magnetic poles. They assume the interaction between electric charges in motion and magnetic fields is something fundamental. Yet Ampere argues that elementary forces must act between objects of the same nature. For example, in Coulomb’s electrostatic law, one describes mutual interaction between electric charges. In Newton’s gravitational law, one describes the gravitational force between two masses. Thus Ampere rejects the idea of interactions between objects of different natures as something fundamental. To illustrate Ampere in his own words, we quote from Ampere [as cited above]:
“1st: Every explanation in the sciences consists in discovering a primitive fact expressed by a general law which, once presented, can be utilized in order to deduce from it all other [facts or laws].”
“2nd: The primitive fact cannot be here the action between a voltaic conductor and a magnet, because, these two things being heterogeneous, the mutual action between them must necessarily be more complicated than that [mutual action] which takes place between two magnets, or than that [mutual action] which I [Ampere] discovered between two conductors”
“Is it not evident that one should look for the primitive fact in the action between two things of the same nature, like two conductors, and not in that action between two heterogeneous things, like a conductor and a magnet?”
As we describe below, this heterogeneity of the Maxwellian electromagnetic fields \(E,B\) is the cause of Einstein’s perceived asymmetry in the Maxwell equations.
Faraday’s Law of Induction
Before we present Einstein’s view on magnetism and special relativity, we briefly review Faraday’s law of induction. From our perspective, we see Faraday’s results on induction as having two parts:
we have voltaic induction, where we have interactions of time-varying electric fields inducing secondary electric fields in conductors. This is an induction between homogeneous elements, namely conductors interacting with conductors.
in application one considers Faraday “magneto” induction, which is the basis behind electrical generators, where mechanical energy is converted into electrical energy.
For example, standard textbooks (Siskind 1959, 1) say: “generator action can take place when, and only when, there is relative motion between conducting wires (usually copper) and magnetic lines of force.” Examples of generators are steam turbines, gasoline engines, electric motors, or hand-powered cranks, etc.. Rotating electrical generators consist of (i) an even set of electromagnets or permanent magnets and (ii) a laminated steel core containing current carrying copper wires (“armature windings”).
“In the DC generator, the armature winding is mechanically rotated through the stationary magnetic fields created by the electromagnets or permanent magnets.”
“In the AC generator, the electromagnets or the permanent magnets and their accompanying magnetic fields are rotated with respect to the stationary armature winding.”
If we suppose the magnetic field \(B\) is properly defined and if \(C\) is a conductor, then Faraday’s law of induction says variations in \(flux(B)\) along the conductor generates a reciprocal emf in the conductor according to the formula: \[\text{emf}:=\int_C E\cdot ds=-\frac{d}{dt} \int_S B \cdot dS=-\int_S \frac{dB}{dt} \cdot dS\] where \(S\) is any surface which bounds the conductor \(\partial S=C\). The induction formula says the circulation of \(E\) along the conductor boundary \(C=\partial S\) is equal to the negative flux of \(\partial{B}/\partial t\) along any bounding surface. We remark that Faraday’s law is invariant whether the magnet is in motion and the conductor at rest, or vice versa, or a combination of both.
Einstein on Maxwell Equations and Magnetism as Relativistic Effect
There is another important influence of magnetism on physics, and this is Einstein’s interpretation of magnetism as a special relativistic effect. An interesting treatment is provided by Prof. J.D. Norton here
There does not appear to be any asymmetry in Faraday’s law of induction. However Einstein’s 1905 paper (Einstein 1905) treatment of magnetism as an effect of special relativity begins from the premise that there is an asymmetry. To paraphrase from the first paragraph: “Maxwell’s equations – as usually understood at the present time – lead to asymmetries which do not appear inherent in the phenomena, for example, in the force interactions of a magnet and conductor in relative motion.”
What is the asymmetry? It’s not easy to precisely locate in the text of Einstein’s paper, but it appears to be this:
- Einstein’s Asymmetry: The heterogeneous magnetic-electric structure of solutions to Maxwell’s equations is not Lorentz invariant. Solutions \((E,B)\) to Maxwell’s equations in an inertial frame \(K\) can satisfy \(E=0\) or \(B=0\), but in another inertial frame \(K'\) the solutions \((E', B')\) to Maxwell’s equations can satisfy \(E'\neq 0\) or \(B' \neq 0\). Equivalently, a solution might be strictly magnetic or electric in inertial frame \(K\) and a combination of electric and magnetic solutions in \(K'\).
Therefore Einstein’s asymmetry consists in the inertial frames \(K, K'\) having different “physical” descriptions of the same event (the induced emf in the conductor). The apparent asymmetry is a consequence of the heterogeneity inherent to Maxwell’s equations, namely the interactions of fields \(E,B\) with distinct physical natures, namely electric versus magnetic. This noninvariance of heterogeneity follows from Einstein’s application of Lorentz transformations to Maxwell’s equations.
Ampere-Weber Explanation of Faraday Magneto Induction
In Ampere’s electrodynamics, magnetic forces are induced by electrical currents in secondary circuits. Therefore in Ampere’s system there is only relational electrical force between current elements. Magnetism in Ampere’s model is explained at the atom physical level by bound electron orbits around heavy central atomic protons. A permament magnet has a lattice structure such that the angular momenta of the electrical atoms are stuctured and have an additive coherent net effect. This means the moments are not randomly aligned but highly correlated in some parallel direction. Therefore if a permament magnet is at rest macroscopically, it’s constituent electrical molecules have dynamic orbits with parallel angular momenta. If the conductor is in motion relative to the permanent magnet, then we have a resultant relative motion of the conductor with all the angular momenta of the constituent electrical particles in the magnet.
Thus in Ampere’s setting, we find voltaic induction (relative motion of conductors) and the magneto induction (relative motion of magnet and conductor) become equivalent. Microscopically there is only induction principle, but macroscopically it manifests somewhat different in permanent magnets.
Where to from here?
Ampere wrote (Assis and Chaib 2015, 29.19):
“Throughout history, whenever hitherto unrelated phenomena have been reduced to a single principle, a period has followed in which many new facts have been discovered, because a new approach in the conception of causes suggests a multitude of new experiments and explanations. It is thus that Volta’s demonstration of the identity of galvanism and electricity was accompanied by the construction of the electric battery with all the discoveries which have sprung from this admirable device….”
In this concluding section, we briefly compare the total Hamiltonian of the heterogeneous standard model of electrons in atoms with the homogeneous Ampere-Weber electrodynamic view. The textbook (Nielsen and Chuang 2010, 7.5.2) briefly presents the total Hamiltonian alleged to describe the electrons in atoms as having the form \[H_{std}=\sum \frac{|p_k|^2}{2m} -\frac{Ze^2}{r_k} +H_{rel}+H_{ee}+H_{so}+H_{hf}, \] where:
the first terms describe the energy balance of kinetic energy with the Coulomb attraction to the positively charged nucleus.
\(H_{rel}\) is a “relativistic correction term”.
\(H_{ee}\) is an electron-electron interaction energy arising from so-called “fermionic” nature of electrons.
\(H_{so}\) is the spin orbit interaction, which is supposed to be the interaction of the electron spin interacting with the magnetic field generated by its motion around the atom.
\(H_{hf}\) is the so=called hyperfine interaction which is supposed to be the electron spin interaction with the magnetic field generated by the nucleus.
The standard Hamiltonian \(H_{std}\) is obviously highly heterogeneous.
But the essence of simplicity is Ampere-Weber’s hypothesis that all atoms are electrical molecules consisting of positive and negative point electric charges. But Weber does not simply take the Coulomb force law, but Weber proposes his fundamental law of electrodynamic interaction represented in the Hamiltonian \[H_{w}=\sum_{ij} \frac{\mu_{ij}}{2} {\nu_{ij}}^2 + \frac{q_i q_j}{r_{ij}} (1-\frac{\nu_{ij}^2}{2c^2}),\] where \(\nu_{ij}:=r'_{ij} = {\hat{r}_{ij}} \cdot v_{ij}\). In otherwords, Weber’s Hamiltonian depends strictly on the scalar relational variables \(r_{ij}, \nu_{ij}\). Weber’s Hamiltonian is the sum of all pairwise interactions in the electrical system, i.e. there is no reference to three-fold or four-fold, etc., interactions.
Our ultimate opinion is that Weber’s Hamiltonian \(H_w\) is the fundamental homogeneous Hamiltonian, while the standard model’s Hamiltonain \(H_{std}\) is heterogeneous. We must imagine that Ampere would insist on Weber’s Hamiltonian being developed as far as possible. This is our goal.
[To be continued –JHM]