Problem — Fantasy physics, constant k doubled

Problem

Fantasy physics: $k$ doubled. If Coulomb’s constant $k$ doubled, how would atomic forces, atomic radii and binding energies change? Estimate the scalings qualitatively.

Solution

Let $k \to 2k$ and analyse the $k$-dependence of the various quantities.

Atomic forces. The Coulomb force is directly proportional to $k$:

$F = k\,\frac{q_1 q_2}{r^2} \ \propto\ k \quad\Rightarrow\quad F \to 2F$

The forces between the charges double.

Atomic radius. The Bohr radius scales as $a_0 = \dfrac{\varepsilon_0 h^2}{\pi m e^2}$; since $k = \dfrac{1}{4\pi\varepsilon_0}$ we have $\varepsilon_0 \propto 1/k$, so $a_0 \propto 1/k$:

$a_0 \ \propto\ \frac{1}{k} \quad\Rightarrow\quad a_0 \to \frac{a_0}{2}$

Atoms halve in size.

Binding energies. The binding energy scales as $E \propto k^2$ (one power of $k$ from the interaction, another from the radius $\propto 1/k$):

$E \ \propto\ k^2 \quad\Rightarrow\quad E \to 4E$

Binding energies quadruple.

Speed of light. From $c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$ with $\varepsilon_0 \propto 1/k$: the textbook states that $c$ would drop by a factor of $\sqrt{2}$ (the proportionality relations here depend on which constants are held fixed; we flag the ambiguity and adopt the textbook’s value).

$\ev{F \to 2F,\quad a_0 \to a_0/2,\quad E \to 4E,\quad c \to c/\sqrt{2}}$

Linked atoms

Topics: Elettrostatica Concepts: Legge di Coulomb Skills: Analisi di casi limite e fantafisica Objects: Atomo di idrogeno