- 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

Galvani (1780s)

Aldini (1800s)

Nijmegen (2000s)

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

- 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

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

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

combine the…

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

and the…

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

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

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

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)

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

Williams, Faller, Hill 1971

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

Revisited by Fulcher (1986):

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

- 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!)

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

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

Suppose that it is. Problem: **field energy diverges**!

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

Suppose it has a radius \(a\).

Good! Energy is finite.

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

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

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?

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