Sci-fi physics: Coulomb 1 over r cubed

Sci-fi physics: Coulomb's law $1/r^3$. Imagine that the Coulomb force decayed as $1/r^3$ instead of $1/r^2$. What would change for: (a) the hydrogen atom, (b) the parallel-plate capacitor, (c) the stability of ordinary matter and the validity of Gauss's theorem?

Solution

Premise. Suppose $F \propto 1/r^3$ (and so the field $E \propto 1/r^3$). What changes is the “steepness” with which the interaction fades with distance: much faster from afar, much stronger up close.

(a) Hydrogen atom — instability. With the $1/r^2$ law the electron finds stable orbits because the attraction decreases “gently”. With $1/r^3$ the force grows too fast as the electron approaches: there is no stable equilibrium between attraction and centrifugal force. An electron pushed slightly inward is attracted more and more and collapses onto the nucleus; one pushed slightly outward escapes. The atom is unstable. (In general stable closed orbits exist only for $1/r$ or harmonic potentials — Bertrand’s theorem.)

(b) Parallel-plate capacitor — non-uniform field. With the ordinary law an infinite charged plane generates a uniform field, independent of distance: this is what makes the parallel-plate capacitor so useful. With $1/r^3$ the contributions from the charge elements of the plane no longer combine into a constant field: the field would depend on the distance from the plates and would no longer be uniform. The linear relation $V = E\,d$ and the simple capacitance formula would break down.

(c) Ordinary matter and Gauss’s theorem. Gauss’s theorem, $\Phi(\vv{E}) = Q_\text{int}/\varepsilon_0$, works precisely thanks to the inverse-square law: only with $E \propto 1/r^2$ does the growth of the spherical surface ($\propto r^2$) exactly compensate the decrease of the field, making the flux independent of the radius. With $E \propto 1/r^3$ the flux through a sphere would become $\propto 1/r$: Gauss’s theorem would no longer hold in its simple form. The pillar that guarantees the arrangement of charge on conductors, shielding and, ultimately, the stability and structure of ordinary matter as we know it would be lost.

Connections

Topics: Electric field and potential Concepts: Coulomb’s law · Gauss’s theorem · Electric field Skills: Applying Gauss’s theorem Methods: Gauss’s theorem Objects: Hydrogen atom