a (re)introduction

Jean-Sébastien Caux

"Classical" electrodynamics is boring because

  • it's a dead subject (it hasn't changed much in the last 100 years)
  • we've studied it before, we know everything already
  • there are much more interesting forces in nature

Interesting historical anecdotes

Galvani (1780s)


Interesting historical anecdotes

Aldini (1800s)


Interesting historical anecdotes

Nijmegen (2000s)


Electrodynamics giving "life"


or death


Energetics of fission reaction: U-235

kinetic energy of neutrons \(\sim 2\) MeV
gamma ray photons \(\sim 7\) MeV
kinetic energy of nuclei \(\sim 170\) MeV

Nuclei fly apart at about 3% of light speed
because of Coulomb repulsion

Important aspects of electric charges and classical electrodynamics

  • charge comes in two different varieties (+ and -)
  • charge is quantized in discrete units
  • charge is conserved (locally)
  • force has inverse square form
  • forces display linear superposition

Elementary charge: how accurately is it known?

Theresa Knott, CC BY-SA 3.0, via Wikimedia Commons

Millikan's oil droplet experiment (1909; Nobel 1923):

\(1.5924(17) \times 10^-19\) C (accurate to 1% of current value)

Most accurate way of measuring \(e\)

combine the…

Josephson effect (voltage to frequency)


\(I(t) = I_c \sin \phi(t)\), \(\frac{d\phi}{dt} = \frac{\hbar}{2e} V(t)\)

and the…

Quantum Hall effect

hall1.png Chang_fig_3.png

Together: \(2 \times \frac{h}{2e} \times \frac{e^2}{h} = e = 1.602176487(40) \times 10^{-19} C\)

How accurately is charge quantized?

Published experiments: \(\frac{|e_{el}| - e_p|}{|e_{el}|} \lesssim 10^{-21}\)

Proposed experiment using cold atoms, lasers and interferometry: \(\frac{|e_{el}| - e_p|}{|e_{el}|} \lesssim 10^{-28}\)

How accurate is Coulomb's law?

Possible deviations:

  • the force is not \(\frac{1}{r^2}\) but \(\frac{1}{r^{2 + \epsilon}}\)
  • the potential has a Yukawa form, \(\frac{e^{-\mu r}}{r}\) where \(\mu = m_\gamma c/\hbar\) ('massive' photon)

Tests of exponents throughout the ages


Cavendish's experiment


See the account of Cavendish's extraordinary work, by none other that J. C. Maxwell, in The electrical researches of the Honourable Henry Cavendish

Most quoted "recent" experiment

Williams, Faller, Hill 1971

\(\epsilon = (2.7 \pm 3.1) \times 10^{-16}\)

Revisited by Fulcher (1986):

\(\epsilon = (1.0 \pm 1.2) \times 10^{-16}\)

Willimans Faller Hill 1971


Tests of the photon mass

  • terrestrial measurements of \(c\) at different frequencies
  • measurement of radio dispersion in pulsar signals
  • lab tests of Coulomb's law
  • limits on constant external magnetic field at Earth's surface (Schrödinger!)

Tests of the photon mass (recent)


Could exist down to a scale given by Heisenberg's uncertainty principle: (age of universe \(\simeq 10^{10}\) years)

\(m_\gamma \simeq \frac{\hbar}{(\Delta t) c^2} \simeq 10^{-66} g\)

Is the electron really a point-like particle?

Suppose that it is. Problem: field energy diverges!

\[ u = \frac{\varepsilon_0}{2} E^2 = \frac{e^2}{32 \pi \varepsilon_0 r^4} \] \[ \rightarrow U = \int dr \frac{e^2}{8 \varepsilon_0 r^2} \rightarrow \infty \]

Is the electron really a point-like particle?

Suppose it has a radius \(a\).

Good! Energy is finite.

Moreover: electromagnetism gives inertia (a mass!) to the electron!

Electromagnetic mass

Suppose that the electron is a shell of negative charge


Accelerated electron displays inertia because of retardation effects of EM self-interaction

Thomson, Heaviside, Lorentz, Poincaré, Fermi, Wilson, etc

What about linear superposition?

When particles come very close to each other:

EM fields become very large!

Maybe the true equations are nonlinear?

Infinite self-energy of point-like electron resolved by nonlinearity?

What about linear superposition?

Famous attempt at classical nonlinear theory of E&M: Born & Infeld (1934)

Actually, quantum electrodynamics provides nonlinear effects


There remain many improperly understood things within electrodynamics

Although remarkably wide, the regimes of validity of the laws of electrodynamics might still be limited


Discovery of a failure of ED would have very interesting consequences:

  • loss of gauge invariance (charge conservation?)
  • speed of light frequency-dependent
  • special relativity not correct as formulated

Enjoy the ride!!