5.2 Resistance and resistivity — Learn

Prerequisites: current, charge and potential difference from Section 5.1; the area of a circle; standard form; the gradient of a straight-line graph.

By the end of this section you can: define resistance and use Ohm's law; describe and compare the $I$ – $V$ characteristics of a resistor, a filament lamp and a diode; use resistivity and $R = \dfrac{\rho L}{A}$ ; and explain how temperature changes the resistance of a metal.

1. Resistance

A component resists the flow of charge through it. The resistance of a component is the potential difference across it divided by the current through it.

R = \frac{V}{I}

The unit of resistance is the ohm, $\Omega$ . One ohm is one volt per ampere. A component of resistance $1\,\Omega$ has a pd of $1\,\text{V}$ when the current through it is $1\,\text{A}$ .

Ohm's law states that for a metallic conductor at constant temperature, the current through it is proportional to the potential difference across it. So the ratio $\dfrac{V}{I}$ is constant, and the resistance does not change.

Worked example 1. A resistor has a potential difference of $6.0\,\text{V}$ across it and a current of $0.40\,\text{A}$ through it. Find its resistance.

R = \frac{V}{I} = \frac{6.0}{0.40} = 15\,\Omega

The resistance is $15\,\Omega$ .

2. Current–potential difference characteristics

An $I$ – $V$ graph shows how the current through a component changes as the potential difference across it changes. The shape of the graph tells you how the component behaves. The resistance at any point is $\dfrac{V}{I}$ , so a steeper line means a smaller resistance.

A fixed resistor. The graph is a straight line through the origin. The current is proportional to the pd, so the resistance is constant. This is the behaviour described by Ohm's law.

A filament lamp. The graph curves. As the pd increases, the current increases, but not in proportion. The filament gets hotter, and a hotter metal filament has a larger resistance. So the graph flattens at higher pds, and the resistance rises as the lamp brightens. At a high pd the filament can glow white, and its resistance is several times its resistance when cold.

A diode. In the forward direction the diode conducts only after the pd reaches about $0.6\,\text{V}$ . Above that the current rises quickly. In the reverse direction the diode blocks the current almost completely, and the current is very small. A diode conducts in one direction only.

Worked example 2. A filament lamp has a pd of $12\,\text{V}$ across it and a current of $0.50\,\text{A}$ through it. Find its resistance. When the lamp is cold, its resistance is about $3.0\,\Omega$ . Explain the difference.

R = \frac{V}{I} = \frac{12}{0.50} = 24\,\Omega

The hot resistance is $24\,\Omega$ . The filament is much hotter when the lamp is lit. A metal conducts better when it is cold, because the ions in the lattice vibrate less and scatter the electrons less often. So the resistance rises as the temperature rises, and the hot resistance is much larger than the cold resistance.

3. Resistivity

The resistance of a wire depends on its shape as well as on the material it is made from. A longer wire has more resistance. A thicker wire has less resistance, because there is more room for the charge to flow.

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

The constant $\rho$ is the resistivity of the material. It is a property of the material, not of the particular wire. Resistivity is measured in ohm metres, $\Omega\,\text{m}$ . The resistivity of a metal is very small, because metals conduct well. The resistivity of an insulator is very large.

Worked example 3. A copper wire has a length of $2.0\,\text{m}$ and a cross-sectional area of $1.0 \times 10^{-6}\,\text{m}^2$ . The resistivity of copper is $1.7 \times 10^{-8}\,\Omega\,\text{m}$ . Find its resistance.

\begin{aligned} R &= \frac{\rho L}{A} \\ &= \frac{1.7 \times 10^{-8} \times 2.0}{1.0 \times 10^{-6}} \\ &= 3.4 \times 10^{-2}\,\Omega \end{aligned}

The resistance is $3.4 \times 10^{-2}\,\Omega$ , or $34\,\text{m}\Omega$ .

Worked example 4. A wire has a resistance of $4.0\,\Omega$ , a length of $1.0\,\text{m}$ and a cross-sectional area of $2.0 \times 10^{-7}\,\text{m}^2$ . Find the resistivity of the material.

\begin{aligned} \rho &= \frac{RA}{L} \\ &= \frac{4.0 \times 2.0 \times 10^{-7}}{1.0} \\ &= 8.0 \times 10^{-7}\,\Omega\,\text{m} \end{aligned}

The resistivity is $8.0 \times 10^{-7}\,\Omega\,\text{m}$ .

Watch out. The resistivity of a material is constant at a fixed temperature, but the resistance of a wire depends on its length and area as well. A short thick copper wire and a long thin copper wire have the same resistivity but very different resistances. Check the question: does it ask for the resistance of a wire, or for the resistivity of the material?

4. Temperature and resistance

The resistance of a metal increases as its temperature rises. The ions in the metal lattice vibrate more, so they scatter the drifting electrons more often. The current is smaller for the same pd, so the resistance is larger.

Some materials behave in the opposite way. A thermistor has a resistance that falls as its temperature rises, so it is useful in a temperature sensor. When a material is cooled below a critical temperature, some materials become superconductors. Their resistance falls to zero, and a current can flow without any energy being wasted as heat.

5. Try it: current–potential difference graphs

The figure draws an $I$ – $V$ graph for one component at a time. Change the component and move the pd slider to see how the current responds.

Predict first. Do not move the controls yet. Write down your answers.

For a fixed resistor, what shape will the graph be?
For a filament lamp, will the graph bend towards the current axis or away from it as the pd rises?
For a diode, what is the current when the pd is reversed?

Test. Choose each component in turn. Move the pd slider from a small value to a large one and watch the current. For the diode, move the slider through zero into the reverse direction.

Explain. For the lamp, explain why the resistance changes as the pd rises. For the diode, explain why the graph is not symmetric. Write two or three sentences and use the idea of charge carriers.

Progress check

Return to the chapter problem. The connecting leads are copper. Their total length is $1.2\,\text{m}$ and their diameter is $2.0\,\text{mm}$ . The resistivity of copper is $1.7 \times 10^{-8}\,\Omega\,\text{m}$ .

Find the cross-sectional area of one lead in $\text{m}^2$ .
Find the resistance of the leads.
The chest and paddles have a resistance of $50\,\Omega$ . Compare the resistance of the leads with this value.

If question 1 goes wrong, halve the diameter to get the radius and use $A = \pi r^2$ . If question 2 goes wrong, use $R = \dfrac{\rho L}{A}$ . If question 3 goes wrong, divide the lead resistance by $50\,\Omega$ and write the result as a percentage.

You should now be able to

define resistance and use $R = \dfrac{V}{I}$ ;
state Ohm's law and say when it applies;
describe and compare the $I$ – $V$ characteristics of a fixed resistor, a filament lamp and a diode;
explain why the resistance of a filament lamp changes as it heats up;
use $R = \dfrac{\rho L}{A}$ to find a resistance, a length, an area or a resistivity;
explain the effect of temperature on the resistance of a metal, a thermistor and a superconductor.