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Solved… riddle of the formula drawn in the snow on a car in Cambridge

Doodles scrawled in the snow on a car windscreen are not an unfamiliar sight when the temperature plummets but, in truth, are rarely worthy of a second glance.

Unless, that is, you’re in Cambridge, where occasionally, instead of daubing an anatomically dubious male appendage in the frost, some clever dick will create something that genuinely exercises the brain.

A formula in the snow on a car windscreen in Cambridge. Picture: Dr Flavio Toxvaerd
A formula in the snow on a car windscreen in Cambridge. Picture: Dr Flavio Toxvaerd

This image was spotted on a car windscreen by Dr Flavio Toxvaerd, a University of Cambridge economist and fellow of Clare college who focuses on industrial organisation and the economics of infectious diseases.

Writing on Twitter, he said: “Where I grew up, kids would write obscenities in the snow. But this is Cambridge. Anyone know what formula it is?”

His tweet generated more than 5,000 likes and more than 400 retweets and more than a few suggestions and solutions (a number of them correct), while others responded with images of more, well, basic snow etchings they’d spotted...

We sought the help of the University of Cambridge’s Cavendish Laboratory, home of its Department of Physics, to confirm the answer to Flavio’s question.

And sure enough, the Cavendish - 30 members of which have won Nobel Prizes since their inception in 1895 - came up with the goods.

“It’s the Lagrangian for electromagnetism,” a scientist at the department told us. “This means that it’s the elegant way to capture all four of the Maxwell equations that describe the electric and magnetic field. Among other things, this is the equation that predicts light!”

The discoveries of James Clerk Maxwell - a 19th-century Scottish mathematician and scientist who studied and worked in Cambridge in two spells - helped bring in a new era in physics. Considered a genius in terms of theory and experimentation, his work provided an essential link between the physics of Newton and Einstein.

He developed the classical theory of electromagnetic radiation - describing electricity, magnetism and light as manifestations of the same phenomenon.

His seminal 1865 work, A Dynamical Theory of the Electromagnetic Field, showed that electric and magnetic fields travel through space as waves, moving at the speed of light. He wrote that "light and magnetism are affections of the same substance" - and bringing together light and electrical phenomena led to him predicting the existence of radio waves.

Maxwell came to the University of Cambridge in October 1850, having already studied at the University of Edinburgh. He attended Peterhouse briefly, but switched before the end of his first term to Trinity, as he thought it would prove easier to obtain a fellowship there. At Trinity, he joined an elite secret debating society known as the Cambridge Apostles.

Having graduated in mathematics with the second highest score in his final examination in 1854 - behind Edward Routh - he was awarded a fellowship earlier than usual in October 1855, but left in November 1856 to take up the role of chair of natural philosophy at Marischal College, Aberdeen, aged just 25.

After narrowly surviving smallpox, he had a productive spell at King's College, London, from 1860-65, before returning to Glenlair in Scotland. But he was in back in Cambridge in 1871, becoming the first Cavendish professor of physics and in charge of the development of the now world-famous Cavendish Laboratory. He designed the original laboratory in Free School Lane and was responsible for the regeneration of physics research in Cambridge.

He died of abdominal cancer on November 5, 1879 at the age of 48.

Today, the Maxwell Centre - which opened on the university’s West Campus on April 7, 2016 - bears his name. It is the focal point for industrial engagement with the physical scientists and engineers on the site.

Maxwell's four equations describe one phenomenon each, but he did not actually create them. Instead, he combined four equations made by Gauss, Faraday and Ampere. He added the displacement current into the Ampere's law - the 4th equation - to complete the equation.

The equations are:

  • Gauss's law for static electric fields
  • Gauss's law for static magnetic fields
  • Faraday's law, which says a magnetic field changing with time produces an electric field
  • Ampere-Maxwell's law, which says an electric field changing with time produces a magnetic field.

Combining the third and fourth equations explains an electromagnetic wave, such as light, that can propagate on its own.

And the combination describes that a changing magnetic field produces a changing electric field, and this changing electric field produces another changing magnetic field. The cycle continues, making an electromagnetic wave that propagates through the space.

The equations can be expressed in integral equations form and in differential equations form.

So the Lagrangian - defined as a function that describes the state of a dynamic system - that was written in the snow on a car in Cambridge is a way of bringing these equations together. And it’s certainly more befitting a city of learning than the usual efforts.

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