How Do You Calculate Resistivity?

For IB Physics SL and HL, calculate resistivity using $\rho = \frac{RA}{L}$, where $R$ is resistance, $A$ is cross-sectional area, and $L$ is conductor length. The SI unit of resistivity is the ohm metre, $\Omega\,\text{m}$.

The Reasoning and Method

The IB data booklet gives the relationship

$$R = \frac{\rho L}{A}$$

Rearranging for resistivity gives

$$\rho = \frac{RA}{L}.$$

This relationship shows that, for a fixed material and temperature, resistance increases with length because charge carriers travel through more material. Resistance decreases when area increases because a wider conductor provides more parallel paths for charge flow. Resistivity itself does not change merely because the wire dimensions change.

For a cylindrical wire of diameter $d$, first calculate its cross-sectional area:

$$A = \pi r^2 = \frac{\pi d^2}{4}.$$

For example, a wire has resistance $4.0\,\Omega$, length $2.0\,\text{m}$, and area $1.5 \times 10^{-6}\,\text{m}^2$. Therefore,

$$\rho = \frac{(4.0)(1.5 \times 10^{-6})}{2.0} = 3.0 \times 10^{-6}\,\Omega\,\text{m}.$$

Dimensional analysis confirms the unit:

$$\frac{\Omega \times \text{m}^2}{\text{m}} = \Omega\,\text{m}.$$

In a practical investigation, determine resistance from $R = \frac{V}{I}$. If several wire lengths are tested while material, temperature, and cross-sectional area remain constant, then

$$R = \left(\frac{\rho}{A}\right)L.$$

A graph of $R$ against $L$ should therefore be linear, with gradient $\frac{\rho}{A}$. Hence, $\rho = A \times \text{gradient}$. A non-zero intercept may indicate contact resistance or resistance in the connecting leads.

In experiments, measure diameter at several positions with a micrometer and average the readings. Since $A \propto d^2$, the percentage uncertainty in area is approximately twice the percentage uncertainty in diameter.

A common misconception is that resistance and resistivity are identical. Resistance depends on length and area, whereas resistivity is a property of the material at a specified temperature; for metallic conductors, temperature changes can alter resistivity.

IB Exam Technique

For a Calculate question, show the equation, substitution, and final answer. Convert all dimensions to SI units before calculating area: for example, $0.50\,\text{mm} = 5.0 \times 10^{-4}\,\text{m}$. Because area depends on the square of diameter, failing to convert before squaring creates a large power-of-ten error. Do not confuse diameter with radius, and include $\Omega\,\text{m}$ in the final result.