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Feynman’s advice to W&M student resonates 45 years later

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    ‘I goofed, and so did you’:  Beulah Cox ’77 holds the 1975 letter from renowned physicist Richard Feynman in which the Nobel Laureate tells Cox that she read his book right, but the book was wrong. Their exchange remains a much-shared anecdote in scientific circles.  Courtesy photo
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Beulah Elizabeth Cox ’77 is an accomplished violinist, a part of the New York City classical music scene for 30 years.

Cox has studied with a Juilliard professor and has a long list of solo, chamber and orchestral credits, including Carnegie Hall, Lincoln Center and solo performances in Paris and London. She’s performed on Broadway, including productions of “Les Misérables” and “Miss Saigon” and as an on-stage violinist in “I Love You. You’re Perfect. Now Change.”

She is the founder and leader of the Ambrosia Trio, a professional ensemble that gave concerts regularly in the New York area and beyond before going on a pandemic hiatus.

Years before making her mark as a professional in the arts, Beulah Elizabeth Cox was a student at William & Mary. 

A wrong answer for the ages

Early in her career as a William & Mary student, Cox turned in a physics exam that contained what became one of the most famous incorrect answers in science.

“Do you know how long ago this was?” Cox asked when a caller wanted to know if she was the Beulah Elizabeth Cox who once corresponded with Richard Feynman.

It was August of 1975 — 45 years ago. Some of the details are lost to the passing years, but here is the gist: Cox saw that she had lost points on a physics exam problem that involved an electrified sphere and Gauss’ Law. Her answer seemed in accord with an explanation she read in Feynman.

But her answer was marked wrong. Perplexed, she wrote to Feynman, holder of any number of science honors, including the 1965 Nobel Prize for Physics. The body of her Aug. 22, 1975 letter read:

I recently took a course in elementary physics at the College of William and Mary in Virginia. An exam question concerned Gauss' Law and conductors, namely, does a hollow conductor shield the region outside the conductor from the effects of a charge placed within the hollow but not touching it?

I read Chapter 5, Volume II of the Feynman Lectures on Physics, and understood all except the next to the last paragraph in which you say “… no static distribution of charges inside a closed conductor can produce any fields outside.” This was confusing, as it seemed to contradict all your previous statements. My instructor showed me how a simple application of Gauss' Law, with a surface of integration enclosing the entire conductor, shows that the E vector outside the conductor is not zero.

Could you perhaps explain what the paragraph in question means? I would greatly appreciate a reply as I am now very confused.

Cox let Feynman know in a postscript that there was more at stake than the satisfying of scientific curiosity:

PS: I must admit I have a devious motive in writing to you because on the exam I answered with the explanation that your book gave. However my instructor did not give me any points, even after I found your book to validate my answer. If you could clarify this question for me I would be very appreciative. Thanking you in advance.

Feynman replied to Cox in a letter dated Sept. 12, 1975:

Your instructor was right not to give you any points, for your answer was wrong, as he demonstrated using Gauss’ law. You should, in science, believe logic and arguments, carefully drawn, and not authorities. You also read the book correctly and understood it. I made a mistake, so the book is wrong. I probably was thinking of a grounded conducting sphere, or else of the fact that moving the charges around in different places inside does not affect things on the outside. I am not sure how I did it, but I goofed. And you goofed, too, for believing me.

The Cox-Feynman correspondence, brief as it was, garnered attention, especially in the internet age: A Google search yields pages and pages of hits. It’s been immortalized in print, too. For starters, the full exchange is contained in Perfectly Reasonable Deviations from the Beaten Track: The Letters of Richard P. Feynman. Kip Thorne also referenced the Cox-Feynman letters prominently in his preface of more than one subsequent edition of The Feynman Lectures on Physics — corrected editions that note that the sphere is grounded.

A copy of Feynman's original letterWhy the reply resonates today

Thorne, the Feynman Professor of Theoretical Physics, Emeritus, at California Institute of Technology and the holder of a 2017 Nobel Prize in Physics, said in an email what he believes makes the Cox-Feynman correspondence so resonant, even after all these years: “Feynman’s straightforward message about not accepting anything on the basis of authority — even if that authority is Feynman himself. It’s a powerful and very important message, and I think it resonates with people who have some sense of the power of critical thinking.”

Feynman was a big name in physics even before his Nobel. Etan Markowitz graduated from Caltech in 1962. He said during his time, Feynman taught freshman and sophomore physics for two years.

“They reserved seats in the lecture hall for enrolled students, because all the upperclass physics students and graduate students all wanted to hear what he had to say,” Markowitz said. “Anyway, Feynman came in every Monday, Wednesday and Friday for two years, talking about whatever he wanted to talk about.”

Markowitz said graduate students set up a tape recorder for each session, and photographed each blackboard Feynman used. This raw material was distilled into The Feynman Lectures. Early iterations of the Lectures contained errata, not surprising considering the volume and variety of material churned out by Feynman.

In fact, Hans von Baeyer, Chancellor Professor of Physics Emeritus at William & Mary, says that Feynman errata is “something of a cottage industry,” and the Cox-Feynman correspondence adds a very human touch to what is mostly a set of corrections, checks and confirmations.

Markowitz even sought to bring Cox herself for a cameo appearance at a Feynman centennial a few years back, but for a number of reasons it didn’t work out.

The text of the Cox-Feynman exchange, perpetuated online and in print, may never be lost. The origin story was harder to pin down. Understandably, Cox didn’t save the exam and 45 years after the fact, can’t recall the Gauss’ Law problem. She also drew a blank on the instructor’s name.

A dive into the microfilm

An appeal to William & Mary Registrar Sallie Marchello brought results. Marchello, “engaging my inner Nancy Drew,” dove into the microfilm and emerged with a number of possible names for instructors. She also found that von Baeyer was department chair at the time. Von Baeyer knew about the letters. In addition, he confirmed that Paul Cisterino was the instructor and provided an email address.

What the instructor remembers

Cisterino, “astounded to be hearing about this,” said in an email that he remembered the incident. He was a graduate student in physics, teaching a summer course for non-majors. Cisterino said he left academia a year or so later for a career in avionics test engineering with the Department of Defense.

“The way I remember it, I gave a take-home test, and that enabled Ms. Cox to consult Feynman,” Cisterino said. “I don't recall exactly what the test question was, but it involved the electric field resulting from a statically charged conducting body.”

He added that the class was a survey course for non-majors, and so he likely would not have required a math-heavy answer, but probably was looking for a general statement about the field, invoking Gauss’ Law.

“She came to me right after I handed back the test and told me her answer was in accord with Feynman,” Cisterino said. “I was baffled by that. Because, Feynman, you know…well, he could do this stuff in his sleep. The principle in question was basic and fundamental, and it would be hard to imagine Feynman making a mistake like that.”

But Feynman did, indeed, make that very mistake. Thorne notes in his preface, “Feynman was uncomfortably aware of this error, and others.” Feynman could have passed the error off as a typo, but he accepted responsibility. More to the point, his “I goofed, and so did you” answer to Cox underscores the fact that students have responsibilities as well.

“Feynman's advice is complicated,” von Baeyer said. “Inside the scientific community his opinion is simple and perfect: Read everything, question everything.

“She acted impeccably and courageously,” von Baeyer added. “But in today's world, it is essential to point out that the moral of the story is NOT that Homer nodded. All scientists, including all Nobel laureates, make mistakes. But that by itself is not enough to dismiss their opinions.”

Feynman’s answer goes public — on the bulletin board

Feynman’s reply had its first public exposure in the form of a copy posted in the corridor of Small Hall shortly after its receipt. “I have a very faint memory of some students, some physics students — graduate students, I believe — pointing at me laughing and giggling and discussing among themselves,” Cox said.

Von Baeyer said that he doesn’t remember Cox personally or the receipt of the letter, and learned of her correspondence years later, from Feynman’s book.

Cisterino said he, like Cox, had a faint memory of the letter being posted publicly for a short time in Small Hall. Cisterino added that he believes the continued circulation of the Cox-Feynman exchange has its roots in the continued interest in Feynman himself. Feynman died in 1988, a public intellectual nearly as prominent in popular culture as he was in physics. Cisterino is among those who consider the response to Cox as “classic Feynman.”

“The ready admission that he goofed — and he doesn’t know why — is so characteristic of the utter baring of his soul,” Cisterino said. “He had no interest in trying to fudge it to save his reputation. Having the grace to say, in essence, ‘beats me; I don’t know what I was thinking’ is so characteristic of his personality as I understand it.”

Cox herself had put the correspondence in the back of her mind, until July 8, 2004, when a letter arrived from a firm of publishers. They were seeking permission to include her letter in a book that turned out to be Perfectly Reasonable Deviations from the Beaten Track.

“This was exciting,” she said. “I told my sister about it. And I said, ‘Was I crazy? What was wrong with me?’ My sister just said, ‘No — that sounds exactly like you.’”

Write to a Nobel Laureate? No big deal.

In fact, Cox confessed that she is a little nonplussed by references to her courage in writing to a Nobel Laureate. She said she never had any trouble initiating contact with authority figures when she had some business to discuss. “We’re all just people,” she says.

She was raised in Hampton and at one time during her youth, family financial constraints put an end to her violin lessons. Cox called up Ronald Marshall, a prominent figure in Hampton Roads classical circles. She kept her voice low, she said, so her family wouldn’t hear her negotiating for violin lessons with the concertmaster of the Peninsula Symphony.

Cox credits her father, the late Willis Franklin Cox, with inoculating her against awe of prominent persons. Willis Cox served as vice mayor of the city of Hampton and was a friend of former Virginia Governor Linwood Holton. Holton had two daughters roughly the same age as Beulah and her sister. The four girls became friends and would happily engage in a bit of gubernatorial schtick.

“The limousine would pull up and the people would expect the governor to get out,” Cox said. “Then the door would open and out would pop four little girls.”

She added that one of her fellow members of the limo quartet was the governor’s daughter Anne Holton, who grew up to serve as Virginia secretary of education and to be interim president of George Mason University. In addition, she is married to U.S. Senator Tim Kaine.

Cox’s comfort in dealing with accomplished and famous people has served her well. She moves easily among the higher strata of the music world. And, like many others, she has noticed the close connection between music and science. She has met many physicians and scientists who are also accomplished musicians. She’s even worked with a doctor’s orchestra in New York City — “And these guys are good!” Cox said.

The fateful take-home exam is no more, but the collected papers of Richard Feynman are held in nearly 100 boxes in an archive at Caltech. He was a productive contributor to — and commentator on — science and he was an equally prolific letter writer. The Beulah Cox correspondence is filed with the other “C’s” (including Francis Crick), in Box 24.

Box 24’s contents hold many testimonies to the contributions of Richard Feynman, but they also speak to the qualities of Beulah Cox, her physics instructor and her alma mater.

“It speaks very well for William & Mary, not only because Beulah was a smart, brave and persistent student, but also because Paul, the instructor, was sophisticated enough to steer his students toward Feynman,” von Baeyer said. “He was also smart enough to do the right and honest thing by grading her correctly, instead of buckling under the incredible authority of Feynman's book.”

Her incorrect answer on the Gauss’ Law problem did not discourage Cox from continuing studies in science. She augmented her music major with other science courses at William & Mary, including Physics of Music, unsurprisingly.

“I've always loved science. I've always loved the way scientists think —the way they question,” she said. “I love that. And I loved Feynman’s answer to me. It's funny, I didn't feel like it vindicated my professor. I felt like it vindicated me.”