Coulomb and Newton: two twins with an asymmetry

Reflection — Coulomb and Newton, twins?

At first glance, Coulomb’s law and the law of universal gravitation are indistinguishable: both are “central forces” that decrease as $1/r^2$, both proportional to the product of two “charges” (mass for gravity, electric charge for Coulomb), both admit a potential that vanishes at infinity and define a conservative vector field. The resemblance is striking — yet, on closer inspection, deep differences emerge.

The first difference is in the numbers. The ratio between the Coulomb force and the gravitational force between two electrons is about $4{.}2\times 10^{42}$: forty-two orders of magnitude. The constant $k \approx 9\cdot 10^9$ is enormously larger than $G$. And yet the matter around us, on a large scale, is held together by gravity, while electricity seems to play a secondary role in planetary motion. Why, if it is so weak?

The answer lies in the signs. Mass is of a single sign and gravity always attracts: it cannot be screened, it does not cancel out, and this is why it dominates dynamics on the astronomical scale despite being extremely weak at the microscopic level. Electric charge, on the other hand, exists in two varieties, positive and negative, and spontaneously tends to neutralise itself: ordinary matter is electrically almost neutral at all scales above the nanometre, and its Coulomb effects screen one another. A star of solar mass, seen from afar, is electrically neutral but gravitationally massive.

Winning at different scales

Gravity wins on large scales (galaxies, stars) because it cannot be screened. Electric forces win on small scales (atoms, molecules) because they are about $10^{40}$ times stronger — but the overall neutrality of matter makes them invisible from afar.

This charge/mass asymmetry is one of the fundamental choices with which nature designed the universe. With three different electric charges (instead of two), chemistry would not work; with mass of both signs, galaxies would dissolve. Modern physics recognises in this balance one of the prerequisites for the very existence of stable structures such as atoms, molecules and macroscopic bodies.

Links

Topics: Electrostatics Concepts: Coulomb’s law · Law of universal gravitation Skills: Analysis of limiting cases and “sci-fi physics”

Related exercises: Worked exercise — Three charges at the vertices of a triangle · Problem — Force between two 1-microcoulomb charges · Problem — Force between two spheres with the same charge