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Spec mapping: OCR H556 Module 2.1 — Physical quantities and units, specifically SI prefixes from pico (10−12) through tera (1012), and the manipulation of prefixed and compound units (km h⁻¹ → m s⁻¹, cm² → m², cm³ → m³). (Refer to the official OCR H556 specification document for exact wording.)
Physics deals with quantities of extraordinary range. The radius of a proton is of order 10−15m. The radius of the observable universe is of order 1026m. That is forty-one orders of magnitude between the smallest and largest lengths an A-Level physicist must routinely handle. The natural unit (the metre) is sized for human bodies and is awkward for both ends of that range. Writing the proton radius as "0.000000000000001 m" is unreadable; writing 1fm ("femtometre") is instantly meaningful.
The remedy is the family of SI prefixes — a coherent ladder of powers-of-ten multipliers that attach to any unit symbol. This lesson covers every prefix you need at OCR A-Level (pico through tera), develops the discipline of converting prefixed units to base SI before any algebra, and trains the awkward cases — area and volume conversions, compound unit conversions like km h−1→m s−1, and densities expressed in g cm−3 instead of the SI kg m−3.
Key principle: When you encounter a prefixed quantity in any physics calculation, the first move — always, before any algebra — is to convert it to base SI units. Almost every numerical error at A-Level is rooted in a forgotten or mis-applied prefix.
Consider the statement "a typical capacitor in a radio tuner has a capacitance of 0.000000000047 farads". It is unreadable. Counting the zeros takes longer than computing whatever you wanted the capacitance for. With a prefix it becomes C=47pF — picofarads — which is instantly parseable and 30 times shorter to write. Prefixes make physics readable, communicable and robust against transcription errors.
The crucial skill, however, is not just recognising prefixes but converting them to base units before doing any algebra. The single biggest source of numerical mark-loss across the H556 papers is failure to convert. Pencil-and-paper or in your head, the move is the same: write the prefixed quantity as a number times 10n times the base unit, with n taken from the prefix table below.
You must memorise the following ten prefixes — symbol, multiplier, and approximate physical context — well enough to recall them under time pressure.
| Prefix | Symbol | Multiplier | Typical use |
|---|---|---|---|
| tera | T | 1012 | data storage (TB), high-energy physics (TeV) |
| giga | G | 109 | radio frequencies (GHz), nuclear binding energies (GeV) |
| mega | M | 106 | resistors (MΩ), power output (MW) |
| kilo | k | 103 | mass (kg), distance (km), force (kN) |
| (none) | — | 100 | base unit |
| centi | c | 10−2 | everyday lengths (cm) |
| milli | m | 10−3 | currents (mA), times (ms), masses (mg) |
| micro | μ | 10−6 | capacitance (μF), small lengths (μm) |
| nano | n | 10−9 | wavelengths (nm), signal timing (ns) |
| pico | p | 10−12 | small capacitance (pF), very short times (ps) |
What does each of the following equal in base units?
(a) 2.5km (b) 47μF (c) 3.2nm (d) 8.0MΩ (e) 650nm (wavelength of red light) (f) 4.0pF
Solutions.
(a) 2.5×103m=2500m (b) 47×10−6F=4.7×10−5F (c) 3.2×10−9m (d) 8.0×106Ω (e) 650×10−9m=6.5×10−7m (f) 4.0×10−12F
Three rules cover every case.
Multiply by the prefix multiplier.
Divide by the prefix multiplier.
Go via the base unit. Never convert prefix-to-prefix in one move — it almost always introduces an error.
To convert 350mm to μm:
350mm=350×10−3m=0.350m=0.350×106μm=3.5×105μm
Exam tip: When in doubt, always go to base units first, do the calculation there, then convert back to whatever prefix the question demands. This is slightly slower but dramatically safer.
A capacitor of capacitance 220μF is charged to a potential difference of 12V. Calculate the charge stored.
Solution. Q=CV. Convert C to base units: C=220×10−6F=2.2×10−4F. Substitute:
Q=2.2×10−4×12=2.64×10−3C=2.6mC(2 s.f.)
A student who substitutes "220" directly would obtain Q=2640C — the charge transferred by a household lightning strike. The factor-of-106 error is invisible to the calculator and only flagged by the physically absurd magnitude. Always convert prefixes before substitution.
When a prefix attaches to a unit that is squared (areas) or cubed (volumes), the prefix multiplier itself is squared or cubed. This is the single most-fluffed conversion at A-Level.
A rectangular silicon chip is 2.0mm×5.0mm. Calculate its area in m2.
Solution. Convert first:
2.0mm=2.0×10−3m,5.0mm=5.0×10−3m
Multiply:
A=(2.0×10−3)×(5.0×10−3)=1.0×10−5m2
Notice that when two millimetres multiply you pick up (10−3)2=10−6 — not 10−3. This is the trap.
| Linear | Area | Volume |
|---|---|---|
| 1km=103m | 1km2=106m2 | 1km3=109m3 |
| 1cm=10−2m | 1cm2=10−4m2 | 1cm3=10−6m3 |
| 1mm=10−3m | 1mm2=10−6m2 | 1mm3=10−9m3 |
| 1μm=10−6m | 1μm2=10−12m2 | 1μm3=10−18m3 |
Common exam mistake: Writing "1cm2=10−2m2". It is 10−4m2. The exponent on the prefix must be raised to the power of the unit.
A block of aluminium has a mass of 54.0g and a volume of 20.0cm3. Calculate its density in kg m−3.
Solution. Convert mass to kg:
m=54.0×10−3kg=0.0540kg
Convert volume to m3:
V=20.0cm3=20.0×10−6m3=2.00×10−5m3
(Note the 10−6, not 10−2 — the prefix is cubed.)
Compute:
ρ=Vm=2.00×10−50.0540=2700kg m−3
This is the well-known density of aluminium, 2.7×103kg m−3. Notice that the equivalent value in g cm−3 is 2.7 — the factor of 1000 cancels because mass-from-g-to-kg gains 10−3 and volume-from-cm³-to-m³ loses 10−6, net factor 103. Many students remember this as the rule "g cm−3×103=kg m−3".
The trickiest conversions at A-Level involve compound units where two or three sub-conversions must be combined. The method is always the same: replace each piece with its base-SI equivalent, then simplify.
A car travels at 108km h−1. Convert to m s−1.
Solution. Write the speed as a fraction and convert the numerator and denominator separately.
108km h−1=1h108km=3600s108×103m=3600108×1000m s−1=30m s−1
The conversion factor is 1km h−1=(1000/3600)m s−1=(5/18)m s−1, and equivalently 1m s−1=3.6km h−1. Many problems quote speeds in km h⁻¹ (driving, athletics, ballistics); the first move in every calculation is to convert.
Water has density 1.00g cm−3. Convert to SI.
Solution.
1.00g cm−3=1cm31.00g=10−6m310−3kg=103kg m−3
So the SI density of water is 1000kg m−3, a value worth committing to memory.
A digital signal has period T=4.0ns. Calculate its frequency in gigahertz.
Solution. Convert T to base units: T=4.0×10−9s. Compute frequency:
f=T1=4.0×10−91=2.5×108Hz
Convert to GHz:
f=1092.5×108=0.25GHz=250MHz
Some prefixed units look similar but are wildly different. Internalise these distinctions.
| Pair | Factor between them | Why it matters |
|---|---|---|
| mm vs μm | 103 | typical hair vs typical cell |
| mA vs μA | 103 | torch bulb vs nerve impulse |
| kN vs N | 103 | car weight vs apple weight |
| GW vs MW | 103 | national grid vs single power station |
| mm2 vs cm2 | 102 | wire cross-section conventions |
| cm3 vs mL | equal — 1cm3=1mL | chemistry crossover |
| L (litre) vs m3 | 1m3=103L | thermal-physics gas problems |
Exam tip: The litre is not an SI unit but is universally used in chemistry and thermal physics. 1L=10−3m3=1000cm3. The ideal gas law pV=nRT demands V in m3 for SI consistency.
flowchart LR
A[tera 10¹²] --> B[giga 10⁹]
B --> C[mega 10⁶]
C --> D[kilo 10³]
D --> E[base 10⁰]
E --> F[milli 10⁻³]
F --> G[micro 10⁻⁶]
G --> H[nano 10⁻⁹]
H --> I[pico 10⁻¹²]
Each adjacent step is a factor of 103 (apart from the centi rung at 10−2 which sits between base and milli in everyday measurement). To move up the ladder, divide by 103 at each step; to move down, multiply by 103. A conversion that crosses, say, μF→F skips two rungs (106 in total); a conversion that crosses GHz→mHz skips four rungs (1012 in total).
Express 5.0GHz in millihertz.
Solution. From giga (109) to base (100) is a factor of 109. From base to milli (10−3) is another factor of 103. Total 1012.
5.0GHz=5.0×109Hz=5.0×1012mHz
A twelve-order-of-magnitude conversion done cleanly in one step.
When under exam pressure, train yourself to immediately convert any prefixed quantity in a question stem to scientific notation the moment you read it. Annotate the question paper. "A resistance of 2.2kΩ" → write "2.2×103Ω" alongside.
This forestalls the most common exam slip: thinking "I'll remember to convert that" and then forgetting. Examiners' reports from across the H556 papers consistently identify failure to convert prefixes as one of the top three sources of avoidable mark loss in numerical questions.
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