Dimensional scaffolding of electromagnetism using geometric algebra

Abstract

Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like divergence, displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green’s functions and, with this, it is possible to obtain retarded Liénard–Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. This anticipates the same results in the real three-dimensional world, but endowed in this case with a nature which is totally absent in one or three dimensions. Electromagnetic waves propagating exclusively at the speed of light in vacuum can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism.

New mathematical model based on geometric algebra for physical power flow in theoretical two-dimensional multi-phase power circuits

“…• As a result, it is possible to present some fundamental laws in a synthetic form that can be directly applied to pure two-dimensional lumped parameter circuits with minimal adaptations. Therefore, we claim that our work also makes a contribution towards the unification of circuit theory and field analysis approach going a step further in our previous investigations 16 . • A relevant novelty is to consider the magnetic field as a bivector (rather than an axial vector).…”

“…The other one is the geometric product, a bilinear operation that results in a multivector, which is a generalization of the vector concept. Armed with these tools, it is possible to waive the cross product and develop a 2D formulation of electromagnetism 16 . GA encompasses and generalizes many of the different mathematical tools and concepts used in various fields, including differential forms, quaternions, complex numbers, tensors or matrices.…”

“…Thus, Maxwell’s equations reduce to a single expression (see Appendix B) where is the electromagnetic field multivector and is the sum of the density current vector ( ) and free charges ( ). As stated in 16 , it can be expanded to the following where is the magnetic field. Note that we use the letter for the electric field instead of the traditional because, in the rest of this document, lowercase bold letters are reserved for vectors while uppercase bold letters are used for multivectors.…”

Light propagation in ( 2+1 )-dimensional electrodynamics: the case of nonlinear constitutive laws

“…with Ẋa ≡ X a ,b t b and ρ = −J a t a . Several interesting features of these equations and their main distinctions from the 3 + 1 case can be found in [13,[18][19][20][21]. From now on, we shall focus on the limit of geometric optics.…”

Current survey of Clifford geometric algebra applications

“…Using GA and GC 172 expresses electromagnetic magnitudes and relations in a gradual way for space dimensions $$$ n=1,2 $$$ and 3 and time. For $$$ n=1 $$$, electric field and scalar and vector potentials get geometric interpretation in spacetime diagrams.…”