Problem — Quantisation of charge

Problem

The charge of an electron is $e = 1{,}602\cdot 10^{-19}\ \text{C}$, and every charge measured in nature is an integer multiple of $e$. Explain the physical meaning of charge quantisation.

Solution

Statement. Electric charge is not a continuous quantity: every observable charge is an integer multiple of the elementary charge $e$. Letting $n$ be the number of excess or deficit elementary charges,

$q = n\,e, \qquad n\in\Z$

Physical meaning. Matter is made of particles (protons and electrons) carrying charge $+e$ and $-e$. Charging a body means adding or removing a whole number of electrons: you cannot transfer “half an electron”. Charge therefore grows in steps of size $e$, never with intermediate values.

Why it looks continuous at the macroscopic scale. The quantum $e = 1{,}6\cdot 10^{-19}\ \text{C}$ is tiny. An ordinary charge of $1\ \mu\text{C}$ corresponds to

$n = \frac{q}{e} = \frac{1{,}0\cdot 10^{-6}}{1{,}6\cdot 10^{-19}} \approx 6\cdot 10^{12}$

that is, thousands of billions of electrons. With such large numbers the unit step is imperceptible and the charge appears to vary continuously, just as the water of a river appears continuous despite being made of molecules.

Fractional charges. No free fractional charges are observed in nature: quarks, which would carry charge $\pm\tfrac{1}{3}e$ or $\pm\tfrac{2}{3}e$, are confined inside protons and neutrons and never appear in isolation. Free particles always have charge an integer multiple of $e$.

$\ev{q = n\,e,\ \ n\in\Z}$

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Topics: Electrostatics Concepts: Electric charge