Why is the ampere a base unit and not the coulomb?

Current can be defined and measured more directly, so the SI system takes it as fundamental and derives charge from Q = It.

What does homogeneous mean?

Every term in the equation reduces to the same base units.

Can a homogeneous equation be wrong?

Yes. Pure numbers such as 12 or have no units, so a missing or incorrect constant passes the check.

Is the kilogram or the gram the base unit?

The kilogram — the only base unit with a prefix built into its name.

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1Working as a PhysicistFoundation

SI Units and Homogeneity

Every measurement in physics is a number and a unit, and the unit matters as much as the number. The $\textbf{SI system}$ builds every unit from just seven base units, which gives you a powerful tool: if an equation’s units do not balance, the equation is wrong.

25 min Topic 1 · Working as a Physicist

What you'll be able to do

Recall the SI base quantities and their units
Express derived units in terms of base units
Use prefixes from pico to tera, and standard form
Check the homogeneity of physical equations

Base quantities and units

The SI system defines seven $\textbf{base quantities}$ . Six appear in A-Level physics: mass (kilogram, kg), length (metre, m), time (second, s), electric current (ampere, A), temperature (kelvin, K) and amount of substance (mole, mol). The seventh is luminous intensity (candela).

Every other quantity is $\textbf{derived}$ from these through a defining equation. Its unit follows from that equation.

Charge is not a base quantity — current is. The coulomb is derived from $Q = It$ , so $1\ \mathrm{C} = 1\ \mathrm{A\,s}$ .

Tip — A frequent trick question asks which of a list is a base unit. The coulomb, newton, volt and joule are all derived.

Derived units

To find the base units of a derived quantity, write its defining equation and substitute the base units of each term.

Force from $F = ma$ : $\mathrm{kg\,m\,s^{-2}}$ , named the newton. Energy from $W = Fs$ : $\mathrm{kg\,m^{2}\,s^{-2}}$ , named the joule. Power from $P = W/t$ : $\mathrm{kg\,m^{2}\,s^{-3}}$ , the watt.

Electrical units need the ampere. Potential difference is energy per charge, $V = W/Q$ , so the volt is $\mathrm{kg\,m^{2}\,s^{-3}\,A^{-1}}$ .

Example 1Express the ohm in SI base units.

R = V/I

2Volt

= \mathrm{kg\,m^{2}\,s^{-3}\,A^{-1}}

3Divide by ampere.

Answer

1\ \Omega = 1\ \mathrm{kg\,m^{2}\,s^{-3}\,A^{-2}}

Prefixes and standard form

Prefixes scale units by powers of ten: pico (p) $10^{-12}$ , nano (n) $10^{-9}$ , micro ( $\mu$ ) $10^{-6}$ , milli (m) $10^{-3}$ , centi (c) $10^{-2}$ , kilo (k) $10^{3}$ , mega (M) $10^{6}$ , giga (G) $10^{9}$ , tera (T) $10^{12}$ .

Convert everything to base SI units before substituting into equations. A wire of cross-sectional area $0.50\ \mathrm{mm^{2}}$ has area $0.50\times10^{-6}\ \mathrm{m^{2}}$ — the prefix is squared along with the unit.

Standard form, $a\times10^{n}$ with $1 \le a < 10$ , makes very large or small values manageable and shows significant figures clearly.

Example 1Convert

2.5\ \mathrm{cm^{3}}

\mathrm{m^{3}}

1\ \mathrm{cm} = 10^{-2}

m, so

1\ \mathrm{cm^{3}} = (10^{-2})^{3} = 10^{-6}\ \mathrm{m^{3}}

Answer

2.5\times10^{-6}\ \mathrm{m^{3}}

Tip — The classic slip is converting $\mathrm{mm^{2}}$ with $10^{-3}$ instead of $10^{-6}$ . Cube or square the conversion factor along with the unit.

Homogeneity

An equation is $\textbf{homogeneous}$ if every term has the same base units. Only homogeneous equations can be physically correct.

To check, reduce each term to base units. Terms that are added or subtracted must match each other, as well as matching the other side.

Homogeneity cannot detect errors in pure numbers. $s = ut + at^{2}$ is homogeneous but wrong — the correct equation has $\tfrac12$ , which carries no units.

Example 1Show that

v^{2} = u^{2} + 2as

is homogeneous.

v^{2}

and

u^{2}

(\mathrm{m\,s^{-1}})^{2} = \mathrm{m^{2}\,s^{-2}}

2as

\mathrm{m\,s^{-2}} \times \mathrm{m} = \mathrm{m^{2}\,s^{-2}}

; the 2 has no units.

AnswerEvery term is

\mathrm{m^{2}\,s^{-2}}

, so the equation is homogeneous.

Equation recap

1\ \mathrm{N} = 1\ \mathrm{kg\,m\,s^{-2}}

Newton.

1\ \mathrm{J} = 1\ \mathrm{kg\,m^{2}\,s^{-2}}

Joule.

1\ \mathrm{V} = 1\ \mathrm{kg\,m^{2}\,s^{-3}\,A^{-1}}

Volt.

Common mistakes to avoid

Calling the coulomb a base unit.

The ampere is the base unit;

1\ \mathrm{C} = 1\ \mathrm{A\,s}

Converting

\mathrm{cm^{2}}

\mathrm{m^{2}}

with

10^{-2}

The factor is

(10^{-2})^{2} = 10^{-4}

Saying a homogeneous equation must be correct.

Dimensionless constants can still be wrong.

Using grams in SI calculations.

The SI base unit of mass is the kilogram.

Key takeaways

SI base units: kg, m, s, A, K, mol (and cd).
Derived units follow from defining equations.
Prefixes scale by powers of ten; square or cube them with area and volume units.
Homogeneous equations have matching base units in every term.
Homogeneity is necessary but not sufficient for a correct equation.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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