REFLECTIONS ON MAXWELL'S TREATISE (Invited Paper)

Abstract

Abstract-Over a period of about twenty years, Maxwell's determination and unification of the equations of electricity and magnetism evolved from his first paper on the subject in 1855-56, "On Faraday's Lines of Force," to the publication of his Treatise on Electricity and Magnetism in 1873. Notwithstanding the many historical accounts and textbooks devoted to Maxwell's work, I have not been able to find a reasonably concise, yet definitive summary of the fundamentals of exactly what Maxwell did in his Treatise and how he did it. This paper represents my own attempt to provide such a summary.

Maxwell's Definition of Electric Polarization as Displacement

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After reaffirming that the macroscopic dipolar electromagnetic equations, which today are commonly referred to as Maxwell's equations, are found in Maxwell's Treatise, we explain from his Treatise that Maxwell defined his displacement vector D as the electric polarization and did not introduce in his Treatise or papers the concept of electric polarization P or the associated electricpolarization volume and surface charge densities, −∇ · P andn · P, respectively. With this realization, we show that Maxwell's discussion of surface charge density between volume elements of dielectrics and between dielectrics and conductors becomes understandable and valid within the context of his definition of electric polarization as displacement D. Apparently, this identification of D with electric polarization in Maxwell's work has not been previously pointed out or documented except very briefly in [2].

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“…He derives the static equations first and then generalizes to the time varying equations. Exactly how he does this, distinguishing between mathematically defined fields and measurable fields, is explained in [2].…”

“…One of the important features of Maxwell's Treatise [1], as mentioned in the paper [2], is Maxwell's introduction of the electric displacement vector D without ever defining an electric polarization vector P or assuming that the change in electric displacement in dielectrics from that in free space is caused by electric dipoles. In stark contrast, Maxwell thoroughly analyzes magnetization M (magnetic polarization) in terms of elementary magnetic dipoles produced by hypothetical magnetic charge separation and introduces the constitutive relation B = µ 0 (H + M) with H the primary magnetic field produced by the hypothetical magnetic charge and µ 0 the permeability of free space.…”

“…(In the present paper, all equations are written using modern SI units.) As explained in [2], although Maxwell assumes that there is some mechanism for charge separation that produces the displacement vector, he does not assume that this induced charge separation in a dielectric is caused by electric dipoles within the dielectric that give rise to a polarization vector P or polarization volume and surface charge densities, −∇ · P andn · P, respectively. He limits his description of dielectrics to the linear constitutive relation D = ǫE [1, Art.…”

“…It is well-known that Maxwell referred to ∂D/∂t as displacement current and J + ∂D/∂t, rather than J + ∂P/∂t, as the total current. However, although mentioned in Section 4.1 of [2], it has not been documented with enough detail that Maxwell used the term "electric polarization" synonymously with the term "electric displacement" or just "displacement." In other words, when Maxwell talks about the electric polarization, he is referring to the vector D and not some imaginary vector P that is never defined in his Treatise or papers.…”