SNELL’S LAW

OF

REFRACTION

Nota Bene

As stated in the first paragraph of our web page titled, Refraction Overview:

The purpose of the REFRACTION section is to (a) present to the reader, the underlying rationale for (spheroidal earth) geodesists using (alleged) tropo[spheric] refraction to explain (in many cases) the visibilty of subjects of observation that should otherwise be below the horizon (i.e., below and normal to the observer’s horizon line of sight), and (b) give the benefit of the doubt to the fallacious spheroidal earth paradigm by allowing readers to assume the impact of maximum tropo[spheric] refraction on their field observation calculations (see Introduction to the (Allegedly Spheroidal) Earth Calculators).

Such calculations are of paramount importance in REVEALING THE PLANAR LARGE-SCALE STRUCTURE OF THE EARTH’S SURFACE.

Introduction

Optical (as well as radar) refraction (i.e., the bending of a wave) is caused by a change in the velocity of electromagnetic waves traversing media having different densities and hence, different refractive indices. The following diagram and description of the law of refraction (or Snell’s law), adapted in part from Born and Wolf,1,2 applies to the case of two distinct, uniform media.

Description of Snell’s Law


Normal Refracted wave Uniform medium 2 ( n  2 ) θ t Apparent source Observer Incident Wave Uniform medium 1 ( n 1 ) Real Source s  (i) θ i s  (t) Boundary

Figure 1. The refraction of a plane wave.

As shown in Figure 1 above, an Incident wave (having a velocity \(v\small_{1}\)) passing through the rarer Uniform medium 1 (having a refractive index \(n\small_{1}\)), becomes a Refracted wave (having a lower velocity \(v\small_{2}\)) upon crossing the Boundary into the (optically) denser Uniform medium 2 (having a higher refractive index \(n\small_{2}\)), i.e., \(n\small_{1}<\normalsize{n}\small_{2}\), and hence, the ratio \(n\small_{2}:\normalsize{n}\small_{1}\) (expressed as \(n\small_{12}\)) is greater than \(1\). To an Observer viewing the Refracted wave, the Apparent source is higher than the Real source.

Born and Wolf state:

   Also from [equation] (\(6\)),3 using Maxwell’s relation §1.2 [equation] (\(14\))4 connecting the refractive index and the dielectric constant,
\[\dfrac{\textrm{sin}\,θ_i}{\textrm{sin}{\,⁡θ_t}}=\dfrac{v_1}{v_2}=\sqrt{\frac{\vphantom{-}ε_2 μ_2}{ε_1 μ_1}}\]\[=\frac{n_2}{n_1}=n_{12}.\tag{8}\]
The relation \(\textrm{sin}\, ⁡θ_i /\textrm{sin}\,⁡θ_t= n_2/n_1\), together with the statement that the refracted wave normal \(s^{(t)}\) is in the plane of incidence, constitute the law of refraction (or Snell’s law).

When \(n_2>n_1\), then \(n_{12}>1\), and one says that the second medium is optically denser than the first medium. … 5

The nature of refraction in the nonuniform medium of the tropo[sphere] is addressed on our webpage titled, Tropo[spheric] Refraction.


— FINIS —


To understand the nature of refraction in the nonuniform medium of the tropo[sphere], the reader is advised to proceed to our web page titled, Tropo[spheric] Refraction.


  1. Max Born and Emil Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Seventh (Expanded) Edition (Cambridge: Cambridge University Press, 1999), p. 39. The following notation from Fig. 1.10 (Refraction and reflection of a plane wave. Plane of incidence.) on p. 39 has been correspondingly utilized in Figure 1 above, some of which is represented in the subsequent excerpt from that page concerning Snell’s law: \(s^{(i)}\) (incident wave normal), \(s^{(t)}\) (refracted wave normal), \(θ_i\) (angle of incidence), and \(θ_t\) (angle of transmission or refraction).↩️

  2. Ibid., p. 13, Cf. Fig. 1.3 Illustrating the refraction of a plane wave.↩️

  3. Ibid., p. 39, equation (\(6\)) is as follows:\[\dfrac{\textrm{sin}\,θ_i}{v_1}=\dfrac{\textrm{sin}\,θ_r}{v_1}\]\[=\dfrac{\textrm{sin}\,θ_t}{v_2},\tag{6}\]where \(θ_r\) is the angle of reflection (not shown in Figure 1 above).↩️

  4. Ibid., §1.2 p. 14, equation \((14)\) is as follows: \[n=\sqrt{\vphantom{-}εμ}\tag{14},\]where the (lower case) Greek letters epsilon (\(ε\)) and mu (\(μ\)) refer to the medium’s dielectric constant and magnetic permeability respectively.↩️

  5. Ibid., pp. 39–40.↩️


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