Back-of-the-Envelope Physics
DOI: 10.1063/1.1768677
In science, physicists are unusual in their emphasis on the primacy of magnitudes and their ability to estimate the numerical values of quantities. Physicists are “numerate”—as in “literate”—but not everyone else is. Enrico Fermi is said to have claimed that a physicist should be able to estimate anything, such as the number of barbers in Chicago or the binding energy of the deuteron, within an order of magnitude. Clifford Swartz mentions Fermi’s position in the preface of Back-of-the-Envelope Physics, which presents 104 “back-of-the-envelope” calculations that use physics to make quantitative estimates of interesting properties.
The problems are divided into 10 chapters ranging from “Force and Pressure,” which contains a calculation of the density of nails required for a comfortable fakir bed, to “Particles and Quanta,” in which Swartz outlines a calculation of the intensity of light from a synchrotron light source. Overall, about half of the book’s problems describe estimates made without recourse to references or other textbooks and thus fit my interpretation of Fermi’s dictum. Other problems in the book require handbook values of quantities that either were not part of my mental arsenal or required textbook relations that were not evident to me. For example, Swartz’s essay on ice skating requires detailed knowledge that I did not have for the ice–water phase diagram.
In all, I enjoyed Swartz’s book and found it fun. Solving the problems became a kind of game, and sometimes I believed I could solve them in a simpler and more persuasive manner than how Swartz had done. In a few cases, his approach seemed unconvincing; at times, I thought of variations or additions that might have found a place in his book. But having known him for a half-century, I am sure that he did not write the book to amuse or test me. Swartz, who served as editor of the Physics Teacher for three decades and was the 1987 recipient of the Oersted Medal of the American Association of Physics Teachers, is a teacher—and his book is meant to teach.
From observing incoming graduate students at my university, I have found that the educational system for training young physicists is effective in bringing students to a reasonable level of formal sophistication. Thus, our students can solve difficult, formal problems on qualifying exams. But when we put a set of back-of-the-envelope problems on the exams, the students don’t do as well as I would like. We don’t teach quick magnitude reasoning, but perhaps we should. It is so important to be able to dispose of bad ideas quickly—not only after one has completed long calculations. And if we can estimate magnitudes competently, we can play a significant role in public affairs, an area in which important questions are too often answered incorrectly by the innumerate. Although Swartz does not ask his readers to estimate the amount of environmental tobacco smoke a non-smoker inhales in a smoky bar or the limit that thermal noise places on biological effects of weak power-line magnetic fields, back-of-the-envelope calculations strongly suggest that—contrary to what some scientists in other fields and some politicians claim—those environmental factors cannot affect health.
Hence physicists young and old should follow Fermi and Swartz and hone a competency in back-of-the-envelope physics. In that endeavor, Swartz’s little book can serve as a pleasant tutorial. And if, like me, you have given up all hope of improvement, the book is still fun—even for a physicist.
More about the Authors
Robert K. Adair. Yale University, New Haven, Connecticut, US .
