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SI units and prefixes

Almost every quantity in A-level physics carries a unit, and every unit on the course is built from the same six blocks. Learn the blocks, the prefixes that scale them, and the ratios that come out with no unit at all. Then carry the units through the calculation itself, because bolting them on at the end is where the last mark goes missing.

Year 12AQA 3.1.1CIE 1.1.1, 1.2.1, 1.2.2, 1.2.3, 1.2.4, 15.1.1OCR A 2.1.1(a), 2.1.2(a), 2.1.2(b), 2.1.2(c), 2.1.2(d), 2.1.2(e), 2.1.2(f), 3.2.1(b), 4.2.5(c)EDEXCEL 1, 94

Watch SI units, prefixes and conversions on the InkPhysics YouTube channel

IN THIS TOPIC

  • Name the six SI base quantities used at A-level, and unpack any derived unit back into them.
  • Check an equation for homogeneity by reducing every term to base units, and say what a failed check proves and what a passed one does not.
  • Use your board's SI prefixes with standard form, and convert between units of the same quantity, such as J and eV, or J and kW h.
  • Head a table column and label a graph axis in the quantity / unit form, so the entries are pure numbers.

COMMON MISCONCEPTION

Units are labels you attach once the number is finished.

Six units underneath everything

The SI system rests on seven base units. A-level physics uses six of them: the kilogram for mass, the metre for length, the second for time, the ampere for electric current, the kelvin for temperature and the mole for amount of substance. The seventh, the candela for light intensity, the specification leaves out, and you are not expected to recall formal definitions of any of them.

One of the six catches people out. Amount of substance is a base quantity in its own right, on exactly the footing of mass and length, and its base unit is the mole. A question that asks for the base units of the molar gas constant expects you to treat mol like any other base unit.

The six SI base quantities used in this course, and their unitsmasskglengthmtimescurrentAtemperatureKamountmol
FIG. 1The six SI base quantities used in this course and their units; the SI's seventh, luminous intensity in candela, is not used at A-level. Every other unit in the subject is assembled from these.

Derived units

Everything else is a derived unit, meaning a combination of base units thrown up by an equation. Because F=maF = ma, force is measured in kg m s−2, a combination used so often it earned its own name, the newton. The pascal is a newton spread over a square metre. The joule is a newton pushed through a metre. A named unit is shorthand, never new information.

A derived unit assembled from base units: the newtonkgms⁻²N1 N = 1 kg m s⁻²
FIG. 2One newton is one kilogram metre per second squared. The name is an abbreviation for the base-unit combination, not a separate thing.

A few quantities come out with no unit at all. Divide one length by another and the metres cancel, which is why strain is a bare number, and the same happens to efficiency, refractive index, relative permittivity and every other ratio of two like quantities. The units are not missing, they cancelled, and the correct thing to write on the answer line is the number by itself. An angle in radians is the same trick, an arc length over a radius, which is why the radian is not one of the six.

So units deserve better than being an afterthought. Keep them attached while you calculate and the answer carries the right unit on its own. Arrive at the wrong one and you have caught a slip in the working several lines before the examiner would have.

Checking an equation for homogeneity

An equation says that two quantities are the same thing, so both sides must carry the same units. Reduce every term to SI base units and compare them. An equation whose terms all agree is homogeneous; one whose terms disagree is wrong, and you have proved that without knowing anything about where the equation came from.

The method is three steps. Replace each symbol by its base units. Collect the powers and cancel. Then compare, not only the two sides but every term that is added to another, since adding a length to a time means nothing whatever the numbers are. Pure numbers such as 2, π and ½ carry no units and vanish at the first step.

WORKED EXAMPLE

A formula that fails the check

Under pressure a student writes the period of a simple pendulum as T=2πg/lT = 2\pi\sqrt{g/l}, with l the length of the pendulum. Decide from base units alone whether it can be right.

The left-hand side is a time, so it has to come out in s. On the right, 2π contributes nothing, so everything hangs on what is inside the root.

g is in m s−2 and l is in m, so g/l has base units (m s−2)/m = s−2. Take the square root and the right-hand side is in s−1.

s−1 against s, so the formula is not homogeneous and cannot be right. Turn the fraction over: l/g gives m/(m s−2) = s2, whose root is s. The correct version is T=2πl/gT = 2\pi\sqrt{l/g}, and the check found it with no derivation and no booklet.

Be clear about what a pass is worth, though. The check is a filter, not a proof. It catches an inverted fraction, a wrong power and a quantity in the wrong place, and it is blind to anything without units. Ek=mv2E_{k} = mv^{2} is perfectly homogeneous and still wrong, because the missing ½ leaves no trace. Failing the check proves an equation wrong; passing it only means you may carry on.

GUIDED PRACTICE

A pressure, checked

Pressure has base units kg m−1 s−2. Show that ρgh, with ρ a density, h a depth and g the gravitational field strength, has the same, and say what that does and does not establish.

Show the working

Density is kg m−3, g is m s−2 and h is m. Multiplying gives kg m−3 × m s−2 × m = kg m−1 s−2.

The two match, so ρgh is homogeneous with a pressure. That does not prove the formula, because a stray numerical factor would survive the check untouched. It does rule out every rearrangement that fails, and ρg/h and ρh/g both fail.

Prefixes and standard form

Real quantities span an enormous range, so the SI attaches prefixes that multiply a unit by a power of ten. Each board requires about ten of them, and the eleven here cover all four:

PrefixSymbolMultiplier
teraT101210^{12}
gigaG10910^{9}
megaM10610^{6}
kilok10310^{3}
decid10−110^{−1}
centic10−210^{−2}
millim10−310^{−3}
microμ10−610^{−6}
nanon10−910^{−9}
picop10−1210^{−12}
femtof10−1510^{−15}
The SI prefixes from femto to tera, deci included, on a powers-of-ten linef10⁻¹⁵p10⁻¹²n10⁻⁹µ10⁻⁶m10⁻³c10⁻²d10⁻¹110⁰k10³M10⁶G10⁹T10¹²
FIG. 3The prefixes on a power-of-ten line. Most step in threes; deci and centi are the odd ones out at ten to the minus one and ten to the minus two.

Most sit at multiples of three, which makes them easy to place on the line. Deci at 10−110^{−1} and centi at 10−210^{−2} are the odd ones out, kept alive by the decibel and the centimetre. AQA's list runs down to femto and omits deci; CIE and OCR require deci and stop at pico. Convert prefixed values to standard form in the base unit before any algebra starts. Write 250 μm as 2.5×10−42.5 \times 10^{−4} m and the calculation can no longer trip over the prefix.

Labelling columns and axes

There is one settled convention for the heading of a table column and the label on a graph axis: write the quantity divided by its unit. Not “speed (m s−1)” and not “speed in m/s”, but speed / m s−1. The slash is doing real arithmetic. Dividing a speed by its unit leaves a pure number, and pure numbers are exactly what a column of readings contains, so a 12 in that column means the speed was 12 m s−1.

load / Nextension / mmextension / m
2.01.41.4×10−31.4 \times 10^{−3}
4.02.92.9×10−32.9 \times 10^{−3}
6.04.34.3×10−34.3 \times 10^{−3}
8.05.85.8×10−35.8 \times 10^{−3}

Any prefix goes into the label as well, which is what the middle and right columns are showing. Head a column extension / mm and its entries are bare numbers such as 1.4; head it extension / m and the same readings become 1.4×10−31.4 \times 10^{−3}. Either heading is correct. Writing “1.4 mm” in every cell of a column already headed with the unit is not, and neither is hanging units along a graph axis next to the numbers.

The convention becomes useful the moment you take a gradient. Divide the label on the vertical axis by the label on the horizontal one and the gradient's unit follows ready-made. Plot speed / m s−1 against time / s and the gradient carries (m s−1)/s, which is m s−2, an acceleration. Every table and every graph on this site is labelled this way, so you have been reading the convention all along.

Converting between units of the same quantity

Some quantities come in more than one unit, and two conversions are worth knowing. One electronvolt is the energy gained by a charge of magnitude e moved through a potential difference of 1 V, so 1 eV=1.60×10−191 \text{ eV} = 1.60 \times 10^{−19} J. One kilowatt hour is a kilowatt delivered for an hour, and 1000 W running for 3600 s makes 1 kW h=3.6×1061 \text{ kW h} = 3.6 \times 10^{6} J.

Neither conversion is printed as an equation. The electronic charge e=1.60×10−19e = 1.60 \times 10^{−19} C is, on the constants page, and that number is the eV factor. Knowing where to look saves you memorising it.

Both conversions run the same way. Write the factor as an equality, decide which direction you are travelling, and multiply. Guessing multiply-or-divide is where these marks go.

WORKED EXAMPLE

A density, converted properly

Aluminium has a density of 2.7 g cm−3. Express it in SI units.

Convert each unit separately before touching the number. 1 g = 10−3 kg, and 1 cm3 = (10−2 m)3 = 10−6 m3. The cube reaches the prefix as well, and that is the step that catches people.

So 2.7 g cm−3 = 2.7 × 10−3 kg / 10−6 m3 = 2.7 × 103 kg m−3.

Does it sense-check? A cubic metre of aluminium ought to have a mass of a couple of tonnes, and 2700 kg is about 2.7 tonnes.

GUIDED PRACTICE

An area with a prefix inside

A wire's cross-section is quoted as 0.45 mm2. Write it in m2, remembering what squaring does to the prefix.

Show the working

1 mm = 10−3 m, so 1 mm2 = (10−3 m)2 = 10−6 m2.

0.45 mm2 = 0.45 × 10−6 = 4.5 × 10−7 m2. Writing 0.45 × 10−3 is a slip markers see constantly: the square must reach the prefix as well.

INDEPENDENT PRACTICE

A speed limit in SI

A road sign reads 36 km h−1. Convert it to m s−1, setting out both conversion factors explicitly.

Show the working

36 km h−1 = 36 000 m per 3600 s.

36 000/3600 = 10 m s−1. Kilometres pushed the number up, hours pulled it down, and only writing both factors keeps the directions straight.

ASSESSMENT FOCUS

  • The prefixes are not printed in the data booklet. Learn your board's full list, awkward centi included.
  • Run a half-remembered or freshly rearranged formula past its base units before you trust it. A mismatch proves the formula wrong; a match only licenses you to continue, because a missing ½ or 2π leaves the units untouched.
  • Head a table column and label a graph axis as quantity / unit, as in speed / m s−1, so the entries are pure numbers. OCR examines the convention directly, and every board's practical papers give marks for it.
  • Squared units square the prefix. 1 cm2 is 10−410^{−4} m2, not 10−210^{−2}, so convert the length first and square afterwards.
  • Going from eV to J, multiply by 1.60×10−191.60 \times 10^{−19}. A photon energy that lands at 101910^{19} J went the wrong way.
  • Write the unit on the answer line every single time there is one. A right number with a missing unit routinely drops the final mark, and it is the cheapest mark on the paper to lose. The exceptions are the ratios, strain and efficiency and refractive index among them, which are pure numbers and take no unit.

CHECK YOURSELF

(a) Write 250 μm in metres, in standard form. (b) An electron has 5.0 keV of kinetic energy. How many joules is that?

Show a hint

Take one prefix at a time: micro first, then kilo, then the eV to J factor.

Show the answer

(a) Micro means 10−610^{−6}, so 250 μm = 250×10−6250 \times 10^{−6} m = 2.5×10−42.5 \times 10^{−4} m.

(b) 5.0 keV = 5.0×1035.0 \times 10^{3} eV. Each eV is 1.60×10−191.60 \times 10^{−19} J, so the energy is 5.0×103×1.60×10−195.0 \times 10^{3} \times 1.60 \times 10^{−19} = 8.0×10−168.0 \times 10^{−16} J.

About ten prefixes per board, and the booklet prints none of them.

WORKBOOK

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  • Name the six SI base quantities used at A-level, and unpack any derived unit back into them.
  • Check an equation for homogeneity by reducing every term to base units, and say what a failed check proves and what a passed one does not.
  • Use your board's SI prefixes with standard form, and convert between units of the same quantity, such as J and eV, or J and kW h.
  • Head a table column and label a graph axis in the quantity / unit form, so the entries are pure numbers.

Open the full revision checklist to track your progress across the whole unit.