A diagram, essentially a graph in the form of a right-angled isosceles triangle of consequent dimensions, 1 ,1, sqrt2 is developed and which represents, in dimensionless form, the individual spectrums of electromagnetic radiation in all media of constant index of refraction. Initially, the metric coordinates associated with the construction are rendered dimensionless by employing the Planck frequency, (V), and Planck length, (L), regarded as a wavelength, as scaling factors. Hence, for the diagram shown here the dimensioned spectrums lie in the ranges, 0 to (V) and infinity to (L).
Further, it is shown that the diagram is universal, in that , other diagrams, covering different ranges, but with identical shape may be constructed provided that the product of the scaling factors of frequency and wavelength is equal to the speed of light in vacuo.
Table of Contents
Introduction
The radiation Strouhal number
The structure equation
The Fidler diagram
Epilogue
Objectives and Topics
This work aims to unify the study of the electromagnetic spectrum and the refractive index by introducing a novel, compact, and dimensionless representation known as the Fidler diagram. The research seeks to overcome the traditional separation of these topics in physics textbooks and provides a pedagogical tool to visualize the behavior of electromagnetic radiation in media with constant refractive indices.
- Unification of electromagnetic spectrum and index of refraction
- Development of the dimensionless Fidler diagram
- Application of Planck units as fundamental scaling factors
- Theoretical derivation of the structure equation
- Exploration of refractive index ranges and radiation characteristics
Excerpt from the Book
The Fidler diagram
The structure equation (7) is v/(V) = Sr (L)/l. Differentiating this we get: d[v/(V)]/d[(L)/l] = Sr = tan θ (8).
It is clear from the diagram that, tan θx = y/(x + y). Hence, the index of refraction, rx = cot θx = (x + y)/y = 1 + x/y. Inspection of the diagram shows clearly that x is some proportion of the distance, (1 – y), say, α, where α lies in the range: 0 ≤ α ≤ 1. Hence we may write: rx = cot θx = (x + y)/y = 1 + x/y = 1 + α(1/y – 1) (9).
Further, the wavelength is given by, (L)/lx = x + y = α(1 – y) + y (10).
From elementary geometry it is easy to show that the (y,x) coordinates of the centroid of the diagram are 1/3 and 2/3, respectively. Hence, the index of refraction at the centroid has a magnitude of 2. It follows that the bisector from lower left to upper right which passes through the centroid represents the spectrum of electromagnetic energy in a medium of index of refraction, 2. Neither of the other two bisectors represents a spectrum.
Summary of Chapters
Introduction: This section identifies the common pedagogical split in physics textbooks regarding optics and the electromagnetic spectrum and proposes the Fidler diagram as a unifying visual solution.
The radiation Strouhal number: This chapter introduces the concept of the radiation Strouhal number, defined by frequency, wavelength, and the speed of light, identifying it as the inverse of the refractive index.
The structure equation: This section derives the fundamental, dimensionless equation that serves as the mathematical basis for constructing the Fidler diagram using Planck units.
The Fidler diagram: This chapter presents the geometric construction of the diagram as a right-angled isosceles triangle and analyzes the relationship between the refractive index and the diagram's coordinates.
Epilogue: The concluding remarks reflect on the potential for future research into frequency quantization and the universal applicability of the diagram's structure.
Keywords
Fidler diagram, electromagnetic spectrum, index of refraction, radiation Strouhal number, Planck units, structure equation, frequency, wavelength, speed of light, non-dispersive media, dimensionless representation, Planck frequency, Planck length, quantum physics, optics
Frequently Asked Questions
What is the primary focus of this research?
The work focuses on unifying the representation of the electromagnetic spectrum and the refractive index into a single, cohesive, dimensionless model.
What are the core topics covered in this document?
The document covers fluid dynamics analogies like the Strouhal number, Planck units, the derivation of the structure equation, and the geometric properties of the Fidler diagram.
What is the main objective of the Fidler diagram?
The objective is to provide a pedagogical and universal tool that represents the spectrum of electromagnetic radiation in all media of constant refractive index.
Which scientific method is utilized in this paper?
The author uses a theoretical approach, combining dimensional analysis, geometric construction, and the application of physical constants to derive a dimensionless structure equation.
What does the main body of the work address?
It addresses the mathematical derivation of the structure equation and shows how this equation allows for the creation of a right-angled isosceles triangle that maps refractive indices against frequency and wavelength variables.
Which keywords define this work?
Key terms include Fidler diagram, electromagnetic spectrum, refractive index, radiation Strouhal number, and Planck units.
How does the radiation Strouhal number relate to the index of refraction?
The radiation Strouhal number is mathematically defined as the inverse of the index of refraction, providing a dimensionless ratio for scaling.
Why is the scaling by Planck units considered fundamental?
Scaling by Planck frequency and Planck length is considered fundamental because it relies on the universal constants of nature rather than arbitrary units.
What is the significance of the triangle dimensions in the Fidler diagram?
The 1, 1, √2 dimensions form a right-angled isosceles triangle that allows for the clear geometric representation of different ranges of refractive indices.
What does the author suggest about space and time divisibility?
The author notes that while some quantum theories suggest space may be discrete, there is currently no convincing proof that space and time are not indefinitely divisible for the purposes of this diagram.
- Quote paper
- William Fidler (Author), 2018, The Fidler Diagram. A compact and dimensionless representation of the spectrum of electromagnetic radiation in all media of constant index of refraction, Munich, GRIN Verlag, https://www.grin.com/document/452282
