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Having established in Chapter 1 that sources ($\rho, \mathbf{J}$) give rise to electromagnetic fields, we now turn to the second link in the causal chain: Propagation. Once a field is generated, it takes on a life of its own, traveling through space governed by the properties of the medium rather than the source that created it.
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The causal chain source $\longrightarrow$ field $\longrightarrow$ observable reflects the factorization of physics into three components: excitations (sources), propagation through a medium (determined by materials and boundary conditions), and measuring quantities (observations). The notes within my classical electromagnetism section will follow this structure with the first section modeling sources.
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“Beauty, like order, occurs in many places in this world, but only as a local and temporary fight against the Niagara of increasing entropy.”
— Robert Wiener, The Human Use of Human Beings
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When light interfaces from one media to another, its phase fronts (lines of constant phase) must stay continuous across the interface of the boundaries. This is true of all physical waves. From this fact, we actually derive Snell’s equation.
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When an electromagnetic wave encounters a boundary between two dielectric media, part of the wave is reflected and part is transmitted. Due to energy conservation, the total power carried by the incident wave must equal the sum of the powers carried by the reflected and transmitted waves, assuming there is no absorption. Additionally, Maxwell’s equations require that the tangential components of the electric and magnetic fields remain continuous across the boundary. To describe this behavior quantitatively, I derive the Fresnel equations for s-polarized (transverse electric, TE) light, where the electric field is perpendicular to the plane of incidence.
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“The vast interplanetary and interstellar regions will no longer be regarded as waste places in the universe, which the Creator has not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find them to be already full of this wonderful medium; so full, that no human power can remove it from the smallest portion of space, or produce the slightest flaw in its infinite continuity.”
— James Clerk Maxwell