Order of magnitude
Physical Quantities & Units - OCR A-Level Physics
Key Definition
Order of magnitude
The power of ten closest to the value of a quantity. If a number is written in standard form as $A \times 10^{n}$, its order of magnitude is $10^{n}$. Order-of-magnitude reasoning is used for rough comparisons, sanity checks, and estimation.
The power of ten closest to the value of a quantity. If a number is written in standard form as $A \times 10^{n}$, its order of magnitude is $10^{n}$. Order-of-magnitude reasoning is used for rough comparisons, sanity checks, and estimation.
Order-of-magnitude estimate
A calculation in which every input is rounded to the nearest power of ten before multiplying or dividing. The aim is to get the exponent right, not the leading digits. Comparing two quantities by their orders of magnitude tells you how many factors of ten separate them.
$$\text{If } x = A \times 10^{n} \text{ with } 1 \le A < 10, \quad \text{order of magnitude of } x = 10^{n}$$
- Comparing orders of magnitude: $2.5 \times 10^{7}$ is three orders of magnitude bigger than $7.1 \times 10^{4}$ because $7 - 4 = 3$.
- Mass of a person: $\sim 10^{2} \text{ kg}$ (order of magnitude $10^{2}$).
- Diameter of an atom: $\sim 10^{-10} \text{ m}$.
- Diameter of a nucleus: $\sim 10^{-15} \text{ m}$. The nucleus is five orders of magnitude smaller than the atom.
- Speed of light: $\sim 10^{8} \text{ m s}^{-1}$.
- Speed of sound in air: $\sim 10^{2} \text{ m s}^{-1}$.
- Height of a person: $\sim 10^{0} \text{ m}$ (about $1$ to $2$ metres).
- Mass of a proton: $\sim 10^{-27} \text{ kg}$.
Worked Example [2 marks]
Estimate the density of concrete to one order of magnitude.
Show Solution
1
Estimate volume of a paving slab
$V \approx 5 \text{ cm} \times 50 \text{ cm} \times 50 \text{ cm} \approx 1 \times 10^{-2} \text{ m}^{3}$.
[1]2
Estimate mass and divide
Mass $m \approx 10 \text{ kg}$, so $\rho = m/V \approx 10 / 10^{-2} \approx 10^{3} \text{ kg m}^{-3}$.
[1]Answer
Order of magnitude $\sim 10^{3} \text{ kg m}^{-3}$, which matches the textbook value of about $2.4 \times 10^{3} \text{ kg m}^{-3}$.
Common Mistake
MEDIUM
Wrong: Quoting a number to 3 significant figures in an order-of-magnitude estimate, for example writing $\rho = 2380 \text{ kg m}^{-3}$ as the "estimate".
Right: The point of an estimate is the nearest power of ten. Write $\rho \approx 10^{3} \text{ kg m}^{-3}$ and state which input values you assumed. Stating extra precision wastes time and can lose the "estimate" mark.
Right: The point of an estimate is the nearest power of ten. Write $\rho \approx 10^{3} \text{ kg m}^{-3}$ and state which input values you assumed. Stating extra precision wastes time and can lose the "estimate" mark.
Examiner Tips and Tricks
- OCR loves estimation questions. Learn key orders: atom diameter ($\sim 10^{-10} \text{ m}$), nucleus diameter ($\sim 10^{-15} \text{ m}$), human mass ($\sim 70 \text{ kg}$), walking speed ($\sim 1.5 \text{ m s}^{-1}$).
- Always justify your estimate with a brief calculation, not a single number out of nowhere. State your input assumptions.
- Sanity-check exam answers by orders of magnitude: a current of $10^{6} \text{ A}$ through a household lamp should immediately look wrong.