Electromagnetic quantities in 3-space and the dual Hodge operator

Abstract

In 3-space, polar vectors (such as a force F) must be distinguished from axial vectors (such as a torque C). A more detailed analysis shows that ‘axial vector’ C is an antisymmetric tensor of the second order. In 3-space, its 9 (= 32) components Cij are grouped together in a 3×3 square. In presentations of electromagnetism, taking account of the existence of these two types of vector, two options exist (α and β), vectors E and B being associated either with group DαHαjα or with group DβHβjβ. Option α (developed in the book Global Geometry of Electromagnetic Systems by Baldomir and Hammond) is based on the use of differential forms: magnitudes in m−1 lead to the polar nature of E and Hα while magnitudes in m−2 correspond to B, Dα and jα. The * Hodge operator then becomes necessary allowing, in particular, the forms *E→EH (the H suffix indicating the role of the Hodge operator), to be able to write Dα = ɛ0EH in the case of the vacuum. This operator is also vital for linking B to Hα, and jα to E. In this paper, we defend option β, showing that the choice of Dβ, Hβ, jβ is justified by purely geometrical arguments; no special operator is then required to establish links between E and Dβ, E and jβ, B and Hβ.