3.1.1 - Use of SI Units and Their Prefixes

3.1.1 - Use of SI Units and Their Prefixes

Physics is a quantitative science built on precise measurement. Every measurement requires a number and a unit, and scientists worldwide use the same system of units -- the Systeme International (SI) -- to ensure consistency and clarity. This lesson covers the six SI base units required by the AQA specification, how to derive the units of other quantities, how to use SI prefixes, and how to convert between different units of the same quantity.

1. The SI Base Units

The SI system is founded on a set of base units, each defined for a fundamental physical quantity. These base units are independent of one another and cannot be expressed in terms of each other. For AQA A-Level Physics, you need to know six of the seven SI base units (the candela, the unit of luminous intensity, is excluded from the specification).

SI Base Units

The SI base units are the fundamental units of measurement from which all other units in physics are derived. Each base unit corresponds to a single base quantity.

The six base quantities and their SI units are:

Base QuantitySymbolSI UnitUnit Symbol
Massmmkilogramkg
Lengthllmetrem
Timettseconds
Electric currentIIampereA
TemperatureTTkelvinK
Amount of substancennmolemol

Note that the kilogram is the only base unit that includes a prefix (kilo-) in its name. This is a historical quirk of the SI system. When applying prefixes to mass, they are applied to the gram rather than the kilogram (e.g. 1 mg = 10−310^{-3} g, not 10−310^{-3} kg).

You are not expected to recall the formal definitions of these base quantities, but you must know what each quantity measures and its associated unit.

2. Derived SI Units

All other physical quantities have units that can be expressed as combinations of the base units. These are called derived units. To find the SI base units of any quantity, start from its defining equation and substitute in the base units for each term.

Derived Unit

A derived unit is a unit of measurement that is expressed as a combination of SI base units through multiplication, division, or raising to a power.

Example 1: Force

Force is defined by Newton's second law:

F=maF = ma

The SI unit of mass is kg and the SI unit of acceleration is m s−2\text{m}\,\text{s}^{-2} (since acceleration = change in velocity / time). Therefore:

[F]=kg×m s−2=kg m s−2[F] = \text{kg} \times \text{m}\,\text{s}^{-2} = \text{kg}\,\text{m}\,\text{s}^{-2}

This combination is given the special name newton (N), so 1 N=1 kg m s−21\,\text{N} = 1\,\text{kg}\,\text{m}\,\text{s}^{-2}.

Example 2: Energy

Energy can be found from the work equation:

W=FsW = Fs

where FF is force and ss is displacement. Substituting the base units of force:

[W]=kg m s−2×m=kg m2 s−2[W] = \text{kg}\,\text{m}\,\text{s}^{-2} \times \text{m} = \text{kg}\,\text{m}^{2}\,\text{s}^{-2}

This is given the name joule (J), so 1 J=1 kg m2 s−21\,\text{J} = 1\,\text{kg}\,\text{m}^{2}\,\text{s}^{-2}.

Example 3: Voltage

Voltage is defined as energy per unit charge:

V=EQV = \frac{E}{Q}

Energy has base units kg m2 s−2\text{kg}\,\text{m}^{2}\,\text{s}^{-2}. Charge Q=ItQ = It, so its units are A s\text{A}\,\text{s}. Therefore:

[V]=kg m2 s−2A s=kg m2 s−3 A−1[V] = \frac{\text{kg}\,\text{m}^{2}\,\text{s}^{-2}}{\text{A}\,\text{s}} = \text{kg}\,\text{m}^{2}\,\text{s}^{-3}\,\text{A}^{-1}

This is given the name volt (V).

The technique of breaking a quantity down into base units is extremely useful for checking equations. If the base units on both sides of an equation do not match, the equation must be wrong.

3. SI Prefixes and Standard Form

Very large and very small quantities appear frequently in physics. Rather than writing out many zeros, we use SI prefixes -- standard multipliers attached to the front of a unit. You must know the following prefixes:

PrefixSymbolMultiplierStandard Form
TeraT1 000 000 000 000101210^{12}
GigaG1 000 000 00010910^{9}
MegaM1 000 00010610^{6}
Kilok1 00010310^{3}
Centic0.0110−210^{-2}
Millim0.00110−310^{-3}
Microμ\mu0.000 00110−610^{-6}
Nanon0.000 000 00110−910^{-9}
Picop0.000 000 000 00110−1210^{-12}
Femtof0.000 000 000 000 00110−1510^{-15}

Note that the prefixes are not evenly spaced: there is no prefix for 10−110^{-1} (deci is not required) but centi (10−210^{-2}) is included. From kilo upward and milli downward, the prefixes step in factors of 10310^{3}.

Standard form expresses a number as a value between 1 and 10 multiplied by a power of 10. For example:

  • 0.0000051 s=5.1×10−6 s=5.1 μs0.0000051\,\text{s} = 5.1 \times 10^{-6}\,\text{s} = 5.1\,\mu\text{s}
  • 64 000 m=6.4×104 m=64 km64\,000\,\text{m} = 6.4 \times 10^{4}\,\text{m} = 64\,\text{km}

To convert a prefixed value to its base unit, replace the prefix with its power of ten:

  • 6 pF=6×10−12 F6\,\text{pF} = 6 \times 10^{-12}\,\text{F}
  • 9 GΩ=9×109 Ω9\,\text{G}\Omega = 9 \times 10^{9}\,\Omega
  • 10 μm=10×10−6 m=1.0×10−5 m10\,\mu\text{m} = 10 \times 10^{-6}\,\text{m} = 1.0 \times 10^{-5}\,\text{m}

In semiconductor manufacturing, the features on a microprocessor chip are measured in nanometres. Modern processors have transistor gate lengths of around 3 nm (3×10−93 \times 10^{-9} m), which is roughly 15 atoms wide. Without SI prefixes, expressing and comparing these dimensions would be impractical.

When performing calculations, always convert prefixed values to base SI units before substituting into equations. This avoids errors from mixing incompatible powers of ten.

4. Converting Between Different Units of the Same Quantity

The AQA specification requires you to convert between different units that measure the same physical quantity. Two important conversions you must be able to perform are between joules and electronvolts, and between joules and kilowatt-hours.

Electronvolts and Joules

Electronvolt (eV)

The electronvolt is the energy gained by a single electron when it is accelerated through a potential difference of one volt. 1 eV=1.6×10−19 J1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}.

Electronvolt Conversion

1 eV=1.6×10−19 J1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}

The electronvolt is used in atomic and nuclear physics because the joule is inconveniently large at these scales. To convert:

  • eV to J: multiply by 1.6×10−191.6 \times 10^{-19}
  • J to eV: divide by 1.6×10−191.6 \times 10^{-19}

When a prefix is attached (e.g. MeV, GeV), convert the prefix first, then convert the unit.

Worked example: Convert 76 MeV76\,\text{MeV} to joules.

Step 1 -- Remove the prefix: 76 MeV=76×106 eV76\,\text{MeV} = 76 \times 10^{6}\,\text{eV}

Step 2 -- Convert to joules: 76×106×1.6×10−19=1.216×10−11 J76 \times 10^{6} \times 1.6 \times 10^{-19} = 1.216 \times 10^{-11}\,\text{J}

So 76 MeV=1.2×10−11 J76\,\text{MeV} = 1.2 \times 10^{-11}\,\text{J} (to 2 significant figures).

Kilowatt-hours and Joules

Kilowatt-hour (kWh)

The kilowatt-hour is the energy transferred by a device with a power of one kilowatt operating for one hour.

Kilowatt-hour Conversion

1 kWh=3.6×106 J=3.6 MJ1\,\text{kWh} = 3.6 \times 10^{6}\,\text{J} = 3.6\,\text{MJ}

The derivation is straightforward: 1 kW=1000 W=1000 J s−11\,\text{kW} = 1000\,\text{W} = 1000\,\text{J}\,\text{s}^{-1}, and 1 hour=3600 s1\,\text{hour} = 3600\,\text{s}. Therefore:

1 kWh=1000×3600=3.6×106 J1\,\text{kWh} = 1000 \times 3600 = 3.6 \times 10^{6}\,\text{J}

Electricity bills use kilowatt-hours rather than joules because the joule is far too small a unit for domestic energy consumption. A typical UK household uses around 2900 kWh of electricity per year, which is 1.044×10101.044 \times 10^{10} J -- a number that would be unwieldy on a bill.

The general strategy for any unit conversion is:

  1. Write down the conversion factor between the two units.
  2. Check the direction: are you going from a larger unit to a smaller one (multiply) or smaller to larger (divide)?
  3. Apply any prefix conversions separately.

5. Bringing It All Together

Being fluent with SI units, prefixes, and conversions is not just an isolated skill -- it underpins every calculation you will perform throughout A-Level Physics. Errors in unit conversion are one of the most common causes of lost marks in examinations.

Always convert all values to SI base units before substituting into any equation. This single habit eliminates the vast majority of unit-related errors in physics calculations.

Here is a summary of the key points:

  • There are six SI base units you need to know: kg, m, s, A, K, mol.
  • Derived units are found by combining base units according to the defining equation of the quantity.
  • SI prefixes range from femto (10−1510^{-15}) to tera (101210^{12}) and must be memorised.
  • Standard form writes numbers as a value between 1 and 10 multiplied by a power of 10.
  • You must be able to convert between J and eV, and between J and kWh.