Problem — Familiar orders of magnitude

Problem

Familiar orders of magnitude. Order from smallest to largest: thickness of a hair, diameter of a red blood cell, hydrogen atom, nucleus of a carbon atom, wavelength of visible light. For each, state the order of magnitude in metres.

Solution

Step 1 — Order of magnitude of each length.

• Carbon nucleus. Nuclear diameter $\sim$ a few femtometres: $\ell \approx 10^{-14}\;\text{m}$.
• Hydrogen atom. Atomic diameter $\sim 1\;$ ångström: $\ell \approx 10^{-10}\;\text{m}$.
• Visible light. Wavelength about half a micrometre: $\ell \approx 5\cdot 10^{-7}\;\text{m}$.
• Red blood cell. Diameter about $7\text{–}8\;\mu\text{m}$: $\ell \approx 10^{-5}\;\text{m}$.
• Hair. Thickness about $50\text{–}100\;\mu\text{m}$: $\ell \approx 10^{-4}\;\text{m}$.

Step 2 — Ordering. Comparing the exponents, $-14 < -10 < -7 < -5 < -4$:

$\ev{\text{C nucleus} < \text{H atom} < \text{visible light} < \text{red cell} < \text{hair}}$

Step 3 — Remarks. The nucleus is about $10^4$ times smaller than the atom that contains it: the atom is almost entirely “empty”. Visible light has a wavelength a thousand times greater than the atom, which is why individual atoms are not visible under an optical microscope. A red blood cell and a hair, though microscopic, are nonetheless of the order of $10^{-5}\text{–}10^{-4}\;\text{m}$: $\sim 10^{10}$ times larger than a nucleus.

Linked notes

Topics: Grandezze fisiche e misura Concepts: Notazione scientifica e ordini di grandezza