CCEA A-Level Physics

Magnitude and unit

A physical quantity is something that can be measured, and every one is written as a number multiplied by a unit. "5" on its own says nothing, whereas 5 m, 5 s and 5 kg are three different statements. The number is the magnitude and the unit says what was being counted. A quantity keeps the same meaning whichever unit it is written in, so a length of 2.5 m is also 250 cm or 0.0025 km, and only the number changes with the unit.

Base units

All measurements in physics can be built from a small set of independent base units (the SI system). Six of them are needed in this course.

Every other unit is a derived unit, meaning a combination of base units found from the equation that defines the quantity. The newton, for example, comes from $F = ma$, which has units of mass times acceleration:

Other derived units follow in the same way.

Expressing a unit in base units

To reduce a unit to base units, substitute the units of each quantity in the defining equation and simplify. For the volt, the equation $V = W/Q$ gives joules per coulomb. Replacing the joule by $\mathrm{kg\,m^2\,s^{-2}}$ and the coulomb by $\mathrm{A\,s}$:

The same process tests an equation. Both sides of a correct equation have the same units, which is called being homogeneous. If $v = \sqrt{T/\mu}$ is proposed for the speed of a wave on a string, with tension $T$ in newtons and mass per unit length $\mu$ in $\mathrm{kg\,m^{-1}}$, the right side has units $\sqrt{\mathrm{kg\,m\,s^{-2} \div kg\,m^{-1}}} = \sqrt{\mathrm{m^2\,s^{-2}}} = \mathrm{m\,s^{-1}}$, which is a speed. Homogeneity shows an equation could be right; it cannot show that it is, because numerical factors such as $\tfrac{1}{2}$ and $2\pi$ have no units.

Prefixes

Prefixes scale a unit by a power of ten, which avoids writing long strings of zeros.

Two cases cause most of the errors. A capital M is $10^6$ and a lower-case m is $10^{-3}$, so 5 MW is five million watts and 5 mW is five thousandths of a watt. Prefixes on powered units apply to the whole unit, so $1\ \mathrm{cm^2} = (10^{-2}\ \mathrm{m})^2 = 10^{-4}\ \mathrm{m^2}$, not $10^{-2}\ \mathrm{m^2}$. In the same way $1\ \mathrm{mm^3} = 10^{-9}\ \mathrm{m^3}$.

Standard form

Standard form writes a number as $a \times 10^n$ with $1 \le a < 10$ and $n$ a whole number. It is used for very large and very small values, and it shows at once how many significant figures were given.

• The wavelength of red light, 700 nm, is $700 \times 10^{-9}\ \mathrm{m} = 7.00 \times 10^{-7}\ \mathrm{m}$.
• A frequency of 2.4 GHz is $2.4 \times 10^{9}\ \mathrm{Hz}$.
• A capacitance of $470\ \mu\mathrm{F}$ is $4.70 \times 10^{-4}\ \mathrm{F}$.

When multiplying numbers in standard form, multiply the front numbers and add the powers of ten. When dividing, divide the front numbers and subtract the powers. If the front number ends up outside the range 1 to 10, adjust it and change the power to match.

Converting units in a calculation

Convert every quantity to base SI units (or to a consistent set of prefixed units) before substituting into an equation. An electron accelerated through a potential difference of 2.5 kV gains $qV$ of energy, where $q = 1.6 \times 10^{-19}\ \mathrm{C}$:

Here 2.5 kV had to become $2.5 \times 10^3$ V first. Leaving it as 2.5 would have given an answer a thousand times too small.

Significant figures in answers

An answer cannot be more precise than the data it was built from, so a final value is given to the same number of significant figures as the least precise quantity used (usually two or three in this course). Intermediate values are kept with extra figures and rounded only at the end, to stop rounding errors building up.