Tag Archives: Maxwell equations

Math can be hard even for very smart people

50 McCosh Hall an autumn evening in 1956. The place was packed. Chen Ning Yang was speaking about parity violation. Most of the people there had little idea (including me) of what he did, but wanted to be eyewitnesses to history.. But we knew that what he did was important and likely to win him the Nobel (which happened the following year).

That’s not why Yang is remembered today (even though he’s apparently still alive at 98). Before that he and Robert Mills were trying to generalize Maxwell’s equations of electromagnetism so they would work in quantum mechanics and particle physics. Eventually this led Yang and Mills to develop the theory of nonAbelian gauge fields which pervade physics today.

Yang and James Simons (later the founder of Renaissance technologies and already a world class mathematician — Chern Simons theory) later wound up at Stony Brook. Simons, told him that gauge theory must be related to connections on fiber bundles and pointed him to Steenrod’s The Topology of Fibre Bundles. So he tried to read it and “learned nothing. The language of modern mathematics is too cold and abstract for a physicist.”

Another Yang quote “There are only two kinds of math books: Those you cannot read beyond the first sentence, and those you cannot read beyond the first page.”

So here we have a brilliant man who invented significant mathematics (gauge theory) along with Mills, unable to understand a math book written about the exact same subject (connections on fiber bundles).

The Pleasures of Reading Feynman on Physics – IV

Chemists don’t really need to know much about electromagnetism.  Understand Coulombic forces between charges and you’re pretty much done.   You can use NMR easily without knowing much about magnetism aside from the shielding of the nucleus from a magnetic field by  charge distributions and ring currents. That’s  about it.  Of course, to really understand NMR you need the whole 9 yards.

I wonder how many chemists actually have gone this far.  I certainly haven’t.  Which brings me to volume II of the Feynman Lectures on Physics which contains over 500 pages and is all about electromagnetism.

Trying to learn about relativity told me that the way Einstein got into it was figuring out how to transform Maxwell’s equations correctly (James J. Callahan “The Geometry of Spacetime” pp. 22 – 27).  Using the Galilean transformation (which just adds velocities) an observer moving at constant velocity gets a different set of Maxwell equations, which according to the Galilean principle of relativity (yes Galileo got there first) shouldn’t happen.

Lorentz figured out a mathematical kludge so Maxwell’s equations transformed correctly, but it was just that,  a kludge.  Einstein derived the Lorentz transformation from first principles.

Feynman back in the 60s realized that the entering 18 yearolds had heard of relativity and quantum mechanics.  He didn’t like watching them being turned off to physics by studying how blocks travel down inclined planes for 2 or more years before getting to the good stuff (e. g. relativity, quantum mechanics).  So there is special relativity (no gravity) starting in volume I lecture 15 (p. 138) including all the paradoxes, time dilation length contraction, a very clear explanation fo the Michelson Morley experiment etc. etc.

Which brings me to volume II, which is also crystal clear and contains all the vector calculus (in 3 dimensions anyway) you need to know.  As you probably know, moving charge produces a magnetic field, and a changing magnetic field produces a force on a moving charge.

Well and good but on 144 Feynman asks you to consider 2 situations

  1. A stationary wire carrying a current and a moving charge outside the wire — because the charge is moving, a magnetic force is exerted on it causing the charge to move toward the wire (circle it actually)

2. A stationary charge and a  moving wire carrying a current

Paradox — since the charge isn’t moving there should be no magnetic force on it, so it shouldn’t move.

Then Feynman uses relativity to produce an electric force on the stationary charge so it moves.  (The world does not come equipped with coordinates) and any reference frame you choose should give you the same physics.

He has to use the length (Fitzgerald) contraction of a moving object (relativistic effect #1) and the time dilation of a moving object (relativistic effect #2) to produce  an electric force on the stationary charge.

It’s a tour de force and explains how electricity and magnetism are parts of a larger whole (electromagnetism).  Keep the charge from moving and you see only electric forces, let it move and you see only magnetic forces.  Of course there are reference frames where you see both.