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Physics deals with phenomena that range from the radius of an atomic nucleus (about 10⁻¹⁵ m) to the distance to the edge of the observable universe (about 10²⁶ m) — over forty orders of magnitude. To make these numbers tractable on a page and in a calculator, physicists use scientific notation and SI prefixes. To make rapid checks of plausibility — "is my answer roughly right, or off by a factor of a million?" — they use order-of-magnitude estimation, sometimes called Fermi-style reasoning after Enrico Fermi, who famously estimated the yield of the Trinity nuclear test by watching scraps of paper move in the blast wave.
This lesson covers the SI prefixes you must know, the conventions of scientific notation, the values you are expected to carry in your head, and the technique of building up a defensible order-of-magnitude estimate from sensible component figures.
Spec mapping: This lesson addresses content from AQA 7408 §3.1 on the use of standard SI prefixes (femto- to tera-) and the conversion between them, together with the skill of order-of-magnitude estimation of physical quantities (refer to the official AQA specification document for exact wording).
Synoptic links:
- Particle physics (§3.2): lengths of 10⁻¹⁵ m, masses of 10⁻²⁷ kg and energies in MeV/GeV demand fluent prefix conversion; an exam-paper slip from MeV to eV costs millions of joules.
- Astrophysics options: astronomical distances (AU, ly, pc) are themselves prefix-style conventions; ability to convert between them and SI metres is examined.
- Required practicals: quoting a current in µA or a length in mm requires that data tables and graphs label the prefix explicitly. Many candidates lose marks for axes labelled "/V" when the data is in millivolts.
You are expected to know, recognise and use the following prefixes. Negative-index prefixes (small quantities) and positive-index prefixes (large quantities) appear in equal numbers in AQA papers.
| Prefix | Symbol | Multiplier | Example |
|---|---|---|---|
| tera | T | 10¹² | terawatt (electricity grid) |
| giga | G | 10⁹ | gigahertz (CPU clock speed) |
| mega | M | 10⁶ | megajoule (chemical energy of food) |
| kilo | k | 10³ | kilometre, kilogram |
| hecto | h | 10² | hectopascal (meteorology — beyond spec but recognised) |
| deca | da | 10¹ | rarely used |
| (none) | — | 10⁰ | base unit |
| deci | d | 10⁻¹ | decimetre (rarely in physics) |
| centi | c | 10⁻² | centimetre |
| milli | m | 10⁻³ | millimetre, millivolt |
| micro | µ | 10⁻⁶ | microsecond, micrometre |
| nano | n | 10⁻⁹ | nanometre (visible light wavelengths 400–700 nm) |
| pico | p | 10⁻¹² | picofarad |
| femto | f | 10⁻¹⁵ | femtometre (nuclear radius) |
A common slip is to write M (mega) when you mean m (milli) or vice versa — a factor of 10⁹ error. AQA mark schemes accept only the correct case: M = mega, m = milli. The lower-case Greek mu (µ) is the only correct symbol for micro; some textbooks use "u" for ASCII convenience, but in exam answers you should write µ.
graph LR
F["femto<br/>10⁻¹⁵<br/>nuclear radius"] --> P["pico<br/>10⁻¹²<br/>atomic spacing"]
P --> N["nano<br/>10⁻⁹<br/>visible light"]
N --> MU["micro<br/>10⁻⁶<br/>red blood cell"]
MU --> MI["milli<br/>10⁻³<br/>pin head"]
MI --> BASE["base unit<br/>10⁰"]
BASE --> K["kilo<br/>10³<br/>town distance"]
K --> ME["mega<br/>10⁶<br/>city to city"]
ME --> G["giga<br/>10⁹<br/>Earth–Moon"]
G --> T["tera<br/>10¹²<br/>solar system"]
style BASE fill:#1f4e79,color:#fff
style N fill:#27ae60,color:#fff
style ME fill:#27ae60,color:#fff
The bar runs from 10⁻¹⁵ m (the femtometre, the rough size of a proton) up through everyday human-scale lengths to 10¹² m (the terametre, comparable to inner solar-system distances). Holding this in mind helps you sanity-check the prefix you use.
Scientific notation expresses any number as a × 10ᵇ, where 1 ≤ a < 10 and b is an integer. This convention is universal in physics and is required for any "give your answer to n significant figures" question.
A frequent A-Level error is typing 6.626 × 10⁻³⁴ as "6.626 × 10 ^ −34", which evaluates 10⁻³⁴ correctly but introduces sign-handling slips when the exponent is squared or square-rooted. Use the EXP or ×10ˣ key on a scientific calculator, which enters the exponent directly: 6.626 EXP −34.
Worked example. A 5 nF capacitor is connected to a 12 V battery. What charge is stored?
Q = CV = (5 × 10⁻⁹) × 12 = 6 × 10⁻⁸ C = 60 nC.
Worked example. Convert a wavelength of 632 nm into metres in scientific notation.
632 nm = 632 × 10⁻⁹ m = 6.32 × 10⁻⁷ m.
The "order of magnitude" of a quantity is the nearest power of ten. So 4500 has order of magnitude 10⁴ (it rounds up; 1000 rounds down to 10³ but 5000 is the boundary, and convention is to take 10³·⁵ ≈ 3160 as the cut-off — quantities above 3160 are 10⁴, below are 10³).
Two quantities are of the same order of magnitude if their ratio is between roughly 1/3 and 3. The mass of a person (≈ 70 kg) and the mass of a sheep (≈ 50 kg) are of the same order; the mass of a person and the mass of a car (≈ 1500 kg) differ by one order of magnitude.
You are expected to carry approximate values for common physical quantities. Examiners use these as anchors for estimation questions.
| Quantity | Approximate value | Order of magnitude |
|---|---|---|
| Mass of an electron | 9.1 × 10⁻³¹ kg | 10⁻³⁰ |
| Mass of a proton | 1.7 × 10⁻²⁷ kg | 10⁻²⁷ |
| Diameter of an atom | 1 × 10⁻¹⁰ m | 10⁻¹⁰ |
| Diameter of a nucleus | 1 × 10⁻¹⁵ m | 10⁻¹⁵ |
| Wavelength of visible light | 500 nm | 10⁻⁷ m |
| Mass of a human | 70 kg | 10² |
| Height of a person | 1.7 m | 10⁰ |
| Mass of a car | 1500 kg | 10³ |
| Speed of sound in air | 340 m s⁻¹ | 10² |
| Speed of light | 3.00 × 10⁸ m s⁻¹ | 10⁸ |
| Atmospheric pressure | 1.0 × 10⁵ Pa | 10⁵ |
| Radius of the Earth | 6.4 × 10⁶ m | 10⁷ |
| Mass of the Earth | 6.0 × 10²⁴ kg | 10²⁵ |
| Earth–Sun distance | 1.5 × 10¹¹ m | 10¹¹ |
| Age of the universe | 4.4 × 10¹⁷ s | 10¹⁷ |
A Fermi problem is one in which you are asked to estimate a quantity using only common knowledge plus order-of-magnitude reasoning. The strategy is to break the problem into smaller estimates each of which you can defend, and then combine them.
This is the canonical Fermi problem in undergraduate-style estimation, and it is worth walking through every step in detail so the structure of the reasoning is transparent. The aim is to estimate how many atoms make up an adult human body, using only physical-chemistry facts that you should be able to summon from memory at A-Level.
Step 1 — Set up the input. A typical adult human has a mass of around 70 kg. This is an order-of-magnitude figure: men average closer to 80 kg, women closer to 65 kg, and children much less, but for an estimation problem we anchor on m_body ≈ 70 kg and accept a factor-of-two band on either side.
Step 2 — Identify the dominant constituent. The body is composed mainly of water (about 60% by mass for an average adult), with proteins, fats and bone minerals making up most of the remaining 40%. For an order-of-magnitude estimate it is acceptable — and conventional — to treat the entire body as water. The justification is that the atomic content of protein, fat and bone is dominated by H, C, N and O at similar number densities to water, so treating everything as H₂O changes the final atom count by a factor of order unity, not by an order of magnitude.
Step 3 — Convert mass to moles. The relative molecular mass of water is M(H₂O) = 2(1) + 16 = 18, so 1 mole of water weighs 18 g = 0.018 kg. The number of moles of water in a 70 kg body is therefore:
n = m_body / M(H₂O) = 70 / 0.018 ≈ 3.9 × 10³ mol ≈ 4 × 10³ mol.
So the body contains roughly four thousand moles of water. A useful sanity check: a litre of water has mass 1 kg and contains about 56 mol, so 70 litres should contain about 70 × 56 ≈ 3900 mol — consistent.
Step 4 — Convert moles to molecules. Avogadro's constant is N_A ≈ 6 × 10²³ mol⁻¹. Multiplying:
N(molecules) = n × N_A = (4 × 10³) × (6 × 10²³) ≈ 2.4 × 10²⁷ molecules ≈ 10²⁷·⁵ molecules.
Step 5 — Convert molecules to atoms. Each water molecule contains 3 atoms (2 H + 1 O). So:
N(atoms in body) ≈ 3 × 2.4 × 10²⁷ ≈ 7 × 10²⁷ ≈ 10²⁸ atoms.
Step 6 — Cross-check the order of magnitude. A more careful calculation that includes the protein, fat and mineral content of the body — using the atomic composition tables from biochemistry — gives a slightly higher figure of about 7 × 10²⁷ atoms, dominated by hydrogen (which is most numerous despite being lightest). The water-only estimate above happens to match this final figure closely. The fact that two independent routes give the same order of magnitude is the standard tell that an estimation is robust.
Step 7 — Reasoning about the result. The answer 10²⁸ atoms is roughly 10⁴ times Avogadro's number, which is itself only the count of atoms in a single mole — a paperclip-sized piece of carbon, say. A human body contains ten thousand Avogadro-clusters of atoms, which is a vivid way to feel the scale of N_A and the connection between the everyday and the microscopic.
The wider technique illustrated here — break the problem into mass → moles → molecules → atoms, multiply through, sanity-check — is the algorithmic core of Fermi estimation. Once internalised it can be deployed on almost any "how many X in a Y" problem at A-Level standard.
Mass of a 747 ≈ 4 × 10⁵ kg (including fuel and passengers). Cruising altitude ≈ 10 km = 10⁴ m. g ≈ 10 m s⁻².
Gravitational PE = mgh = (4 × 10⁵)(10)(10⁴) = 4 × 10¹⁰ J ≈ 10¹⁰ J.
For comparison, 1 litre of jet fuel releases about 4 × 10⁷ J on combustion, so lifting the aircraft requires roughly 10¹⁰ / 4 × 10⁷ ≈ 250 litres of fuel — consistent with the fact that take-off and climb account for a significant fraction of a flight's total fuel burn.
A classic Fermi problem. Population of London ≈ 9 × 10⁶. Suppose 1 household in 50 has a piano: that gives 9 × 10⁶ / (50 × 2.5 people per household) ≈ 7 × 10⁴ pianos. Each piano is tuned roughly once per year. A piano tuner can tune perhaps 4 pianos per day, working 200 days per year, so tunes about 800 pianos per year. Number of tuners ≈ 7 × 10⁴ / 800 ≈ 90.
The actual figure (from professional registers) is around 50–100 — the order of magnitude is right. This is the heart of Fermi reasoning: do not aim for precision; aim for the right power of ten.
The reason A-Level physics insists on fluency with prefixes and scientific notation is that the subject spans an extraordinary range of length, mass and energy scales. A student who can hold the broad map of those scales in their head has a vastly more secure grasp of the discipline than one who treats each topic in isolation.
The table below walks through the length-scale ladder from the smallest known structures of matter up to the size of the observable universe. At each rung a one-line worked check confirms that the order of magnitude is sensible.
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