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 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 Lienard-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 entirely at the speed of light 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.

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使用几何代数的电磁量纲脚手架

使用几何代数和微积分来表达电磁定律，我们能够以渐进的方式呈现量级和关系，从而增加维数。在一维情况下，电荷和电流密度、电场 E 以及标量和矢量势在时空图中得到几何解释。应用于这些量级的几何矢量导数产生了简单的表达式，导致了位移电流、连续性和规范或延迟时间等概念，具有明确的几何意义。由于几何矢量导数是可逆的，我们引入了简单的格林函数，从而可以获得以光速自然传播的延迟 Lienard-Wiechert 势。在二维中，这些量级变得更加复杂，磁场 B 表现为一维世界中不存在的伪标量。感应定律反映了 E 和 B 之间的关系，可以得出电容器、电路和坡印廷矢量的概念，解释能量的流动。这个二维场景中波动方程的解现在揭示了物理效应以光速传播。这预期在真实的三维世界中会有相同的结果，但在这种情况下赋予了一种在一维或三维中完全不存在的性质。因此，完全以光速传播的电磁波可以被视为生活在具有奇数空间维度的世界中的结果。最后，