Hot Molecules, Cold Electrons: From the Mathematics of Heat to the Development of the Trans-Atlantic Telegraph Cable

Description

An entertaining mathematical exploration of the heat equation and its role in the triumphant development of the trans-Atlantic telegraph cable Heat, like gravity, shapes nearly every aspect of our world and universe, from how milk dissolves in coffee to how molten planets cool. The heat equation, a cornerstone of modern physics, demystifies such processes, painting a mathematical picture of the way heat diffuses through matter. Presenting the mathematics and history behind the heat equation, Hot Molecules, Cold Electrons tells the remarkable story of how this foundational idea brought about one of the greatest technological advancements of the modern era. Paul Nahin vividly recounts the heat equation’s tremendous influence on society, showing how French mathematical physicist Joseph Fourier discovered, derived, and solved the equation in the early nineteenth century. Nahin then follows Scottish physicist William Thomson, whose further analysis of Fourier’s explorations led to the pioneering trans-Atlantic telegraph cable. This feat of engineering reduced the time it took to send a message across the ocean from weeks to minutes. Readers also learn that Thomson used Fourier’s solutions to calculate the age of the earth, and, in a bit of colorful lore, that writer Charles Dickens relied on the trans-Atlantic cable to save himself from a career-damaging scandal. The book’s mathematical and scientific explorations can be easily understood by anyone with a basic knowledge of high school calculus and physics, and MATLAB code is included to aid readers who would like to solve the heat equation themselves. A testament to the intricate links between mathematics and physics, Hot Molecules, Cold Electrons offers a fascinating glimpse into the relationship between a formative equation and one of the most important developments in the history of human communication. Cover Contents Foreword by Judith V. Grabiner 1. Mathematics and Physics 1.1 Introduction 1.2 Fourier and The Analytical Theory of Heat 1.3 A First Peek into Fourier’s Mathematical Mind 2. Fourier’s Mathematics 2.1 Fourier Series 2.2 Fourier Transforms 2.3 Fourier Transforms and Dirichlet’s Discontinuous Integral 3. The Heat Equation 3.1 Deriving the Heat Equation in an Infinite Mass 3.2 Deriving the Heat Equation in a Sphere 3.3 Deriving the Heat Equation in a Very Long, Radiating Wire 4. Solving the Heat Equation 4.1 The Case of a Semi-Infinite Mass with a Finite Thickness 4.2 The Case of a Cooling Sphere 4.3 The Case of a Semi-Infinite Mass with Infinite Thickness 4.4 The Case of a Circular Ring 4.5 The Case of an Insulated Sphere 5. William Thomson and the Infinitely Long Telegraph Cable Equation 5.1 The Origin of the Atlantic Cable Project 5.2 Some Electrical Physics for Mathematicians 5.3 Derivation of the Heat Equation as the Atlantic Cable Equation 5.4 Solving the Atlantic Cable Equation 6. Epilogue 6.1 What Came after the 1866 Cable 6.2 The Cable Equation, Duhamel’s Integral, and Electronic Computation Appendix: How to Differentiate an Integral Acknowledgments Notes Index Also by Paul J. Nahin