Abstract
A mathematically rigorous, physically based development of the classical theory of electromagnetism is introduced here through a consideration of the microscopic Maxwell–Lorentz theory. Although the Lorentz theory of electrons is a purely classical, heuristic model that is incapable of analyzing many fundamental problems associated with the atomic constituency of matter, it is nevertheless an expedient model in providing the proper source terms for the microscopic Maxwell equations. Indeed, the classical Lorentz theory does yield many results connected with the electromagnetic properties of matter that agree in functional form with that given by quantum theory. In particular, the Lorentz theory assumes additional forces of just the right nature such that qualitatively correct expressions are obtained and, by empirical adjustment of the parameters appearing in these ad hoc force relations, quantitatively correct predictions may also be obtained. Even though quantum theory justifies the assumption of these additional forces and shows them to be of electrical origin, the Lorentz theory is incapable of arriving at this fundamental level of understanding.
“War es ein Gott der diese Gleichungen schrieb?” Ludwig Boltzmann, quoting from Goethe’s Faust, on Maxwell’s equations.
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Notes
- 1.
The adjective “ponderomotive” describes the tendency to produce movement of a ponderable body. By a “ponderable body” is meant one having nonzero mass.
- 2.
Conservation laws in nature originated with Democritus, a fifth century B.C. atomist who asserted that matter was indestructible.
- 3.
This occurs, for example, in the collisional broadening of radiation from a Lorentz atom. See Chap. 12 of Stone [4] for a detailed development of collisional broadening effects in a system of Lorentz atoms.
- 4.
The electron spin is a purely quantum-mechanical effect that is not susceptible to a complete analysis in classical physics. For a detailed historical discussion of this point, see Chap. VI of Kramers’ classic treatise Quantum Mechanics [7].
- 5.
For a complete rigorous development of radiation reaction in the Lorentz–Abraham model, see Yaghjian’s treatise Relativistic Dynamics of a Charged Sphere [8].
- 6.
- 7.
Galilean invariance states that physical phenomena are the same when viewed by two observers moving with a constant velocity V relative to each other, provided that the coordinates in space and time are related by the Galilean transformation r ′ = r + V t, t′ = t.
- 8.
The order symbol \(\mathcal {O}\) is defined as follows. Let f(z) and g(z) be two functions of the complex variable z that possess limits as z → z 0 in some domain \(\mathcal {D}\). Then \(f(z) = \mathcal {O}(g(z))\) as z → z 0 iff there exist positive constants K and δ such that |f(z)|≤ K|g(z)| whenever 0 < |z − z 0| < δ.
- 9.
While it is practically impossible to measure a zero rest mass, upper limits on the rest mass of a photon have been obtained experimentally by Fischbach et al. [33]. Their results provide a geomagnetic limit given by m γ ≤ 1 × 10−48 g, eighteen orders of magnitude smaller than the rest mass m e of an electron.
- 10.
This result was also derived by Oliver Heaviside in the same year.
- 11.
See J. J. Thompson, Recent Researches, pp. 251–387.
- 12.
Leopold Kronecker (1823–1891) introduced this delta function in his 1866 paper “Über bilineare Formen” presented to the Royal Prussian Academy of Sciences in Berlin that was later published in the Monthly Bulletin of the Academy in 1868.
- 13.
For an electrostatic field this force is given by \(\mathcal {T}_{ij}n_jda\) at the surface \(\mathcal {S}\) of a conductor, the sign change accounting for the fact that the unit normal vector \(\hat {\mathbf {n}}\) is directed outwards from the conductor body. Because the electric field is normal on the surface of a perfect conductor, then \(d\mathbf {f}(\mathbf {r})/da = \|\frac {1}{4\pi }\|\frac {1}{2}\epsilon _0e^2\hat {\mathbf {n}} = \frac {\|4\pi \|}{2\epsilon _0}\rho _s^2(\mathbf {r})\hat {\mathbf {n}} = \frac {1}{2}\rho _s(\mathbf {r})\mathbf {e}(\mathbf {r})\) when \(\mathbf {r} \in \mathcal {S}\).
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Oughstun, K.E. (2019). Microscopic Electromagnetics. In: Electromagnetic and Optical Pulse Propagation . Springer Series in Optical Sciences, vol 224. Springer, Cham. https://doi.org/10.1007/978-3-030-20835-6_2
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