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

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. "This book is a testament to the intimate, mutual embrace of mathematics and physics. It achieves that by telling the story of an historical event of tremendous impact upon society, both spiritually and technically - the mid-19th century construction of the trans-Atlantic telegraph cable, which reduced the time to send a message across the ocean from weeks to minutes. The story of the cable actually begins decades earlier, at the start of the century, with the French mathematical physicist Joseph Fourier's development of the mathematics that the Scottish physicist William Thomson (later Lord Kelvin) would use to analyze the electrical physics of the cable. The story of Fourier opens the book, that of Thomson completes it, and in-between the reader will learn how to derive Fourier's second-order partial differential equation for the flow of heat energy in matter, how Fourier solved the heat equation, how Thomson used Fourier's solutions to calculate the age of the Earth (imagined to be the result of the of an initially molten sphere of blinding brilliance) and, finally, how Thomson showed that the heat equation also describes the Atlantic cable. An epilogue describing the post-Thomson developments completes the book. All readers who have completed first courses at the level of AP-calculus and AP-physics will be able to read this book. This is a perhaps surprising feature of the book, as the mathematics discussed is normally not encountered until the second year (or even later) of college-level work. This book shows that, in fact, the technical material is fully graspable by a college freshman. Unlike a pure technical book, readers will also find a lot of fascinating history in this book (including the bizarre story of how the English novelist Charles Dickens used the Atlantic cable to send a coded message - during his 1867 American reading tour - to avoid a career-damaging scandal concerning his mistress)"-- Provided by publisher Cover 1 Contents 10 Foreword by Judith V. Grabiner 12 1. Mathematics and Physics 18 1.1 Introduction 18 1.2 Fourier and The Analytical Theory of Heat 19 1.3 A First Peek into Fourier’s Mathematical Mind 22 2. Fourier’s Mathematics 35 2.1 Fourier Series 35 2.2 Fourier Transforms 60 2.3 Fourier Transforms and Dirichlet’s Discontinuous Integral 65 3. The Heat Equation 74 3.1 Deriving the Heat Equation in an Infinite Mass 74 3.2 Deriving the Heat Equation in a Sphere 80 3.3 Deriving the Heat Equation in a Very Long, Radiating Wire 88 4. Solving the Heat Equation 96 4.1 The Case of a Semi-Infinite Mass with a Finite Thickness 96 4.2 The Case of a Cooling Sphere 106 4.3 The Case of a Semi-Infinite Mass with Infinite Thickness 111 4.4 The Case of a Circular Ring 119 4.5 The Case of an Insulated Sphere 126 5. William Thomson and the Infinitely Long Telegraph Cable Equation 132 5.1 The Origin of the Atlantic Cable Project 132 5.2 Some Electrical Physics for Mathematicians 135 5.3 Derivation of the Heat Equation as the Atlantic Cable Equation 148 5.4 Solving the Atlantic Cable Equation 153 6. Epilogue 159 6.1 What Came after the 1866 Cable 159 6.2 The Cable Equation, Duhamel’s Integral, and Electronic Computation 174 Appendix: How to Differentiate an Integral 200 Acknowledgments 208 Notes 212 Index 224 Also by Paul J. Nahin 230