Maxwell's Definition of Electric Polarization as Displacement

Abstract

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].

Maxwell's Derivation of the Lorentz Force From Faraday's Law

“…783] but his electric force on a stationary unit electric charge is still given by E(r, t) = −∂A(r, t)/∂t − ∇ψ e (r, t), a relationship that holds independently of the gauge and that Maxwell writes down explicitly for time varying fields in Article 783. Although Maxwell explained in Article 599 that the vector called E v (r, t) herein is also the force experienced by the electric-polarization and conduction charges of a material body (which could be the ether) moving with the curve C(t) as confirmed by his writing in Articles 608, 609, and 619 that D = E v (Maxwell looked at D as electric polarization [13]) and J = σE v (correct for v 2 /c 2 1), it seems clear from what Maxwell wrote in Articles 598 and 599 that he fully realized that −∂A(r, t)/∂t − ∇ψ e (r, t) was the force exerted on a hypothetical stationary unit electric charge placed at the point r and that −∂A(r, t)/∂t − ∇ψ e (r, t) + v × B was the force exerted on a hypothetical moving (with velocity v) unit electric charge placed at the point r, even though he did not explicitly write E v (r, t) = E(r, t) + v(r, t) × B(r, t) as is done in Eq. (23).…”

Maxwell's derivation of the Lorentz force from Faraday's law

“…783] but his electric force on a stationary unit charge is still given by E(r, t) = −∂A(r, t)/∂t − ∇ψ e (r, t). Although Maxwell viewed the vector called E v (r, t) herein as a field vector experienced by a body moving with C(t) as evidenced by his writing in Articles 608, 609, and 619 that D = ǫE v (Maxwell looked at D as electric polarization [13]) and J = σE v (correct for v 2 /c 2 ≪ 1), it seems from what Maxwell wrote in Articles 598 and 599 that he realized that −∂A(r, t)/∂t − ∇ψ e (r, t) was the force exerted on a hypothetical stationary unit electric charge placed at the point r and that −∂A(r, t)/∂t − ∇ψ e (r, t) + v × B was the force exerted on a hypothetical moving (with velocity v) unit electric charge placed at the point r, even though he did not explicitly write E v (r, t) = E(r, t) + v(r, t) × B(r, t) as in (23).…”

“…In magnetic polarization (magnetization), he determines his mathematically defined (macroscopic) magnetic fields [B, H] from his primary free-space magnetic field H 0 measured in small cavities. He does not give an analogous prescription for determining the mathematically defined electric field from cavity fields in polarized dielectrics since Maxwell considered D to be the electric polarization and did not introduce a polarization vector P [2],[13] 8. Even though Maxwell derived the force on electric charge moving through electromagnetic fields, apparently most of the scientific community did not understand what he had done until much later when Lorentz used this force extensively in his work 9.…”

Force and Hidden Momentum for Classical Microscopic Dipoles

“…Adding the equations that result by crossing E or D into (8a) and B or H into (8b) to obtain four possible electromagnetic-field momenta, then making use of the constitutive relations in (9), one can obtain an unlimited number of different macroscopic force-momentum density equations depending on the chosen stress dyadic. Restricting 1 Contrary to what is sometimes stated in the historical literature, Maxwell (and not the "Maxwellians") determined all the equations in (8) [15,16] for the mathematically defined fields of an ideal dipolar continuum where the polarization densities are perfectly continuous throughout the medium rather than composed of discrete dipoles as in a macroscopic dipolar continuum [17]. It is unequivocally shown in [17], [18, sec.…”

“…The electric and magnetic dipole moments p and m induced by external fields on PEC's are not isolated, but the force on the electrically small PEC has already been proven[25], as explained above, to be equal to the sum of the electric-and magnetic-dipole forces in(15) and(16).…”