What decides the resistance of a wire

Why is a thin wire a poor conductor?

Think of water that flows through a pipe. A long, thin pipe lets less water pass than a short, wide one. A wire acts in much the same way. We can find what decides the resistance of a wire with a simple test, where we change one thing at a time.

What the resistance depends on

1. The length $L$. A wire that is twice as long has twice the resistance. $R \propto L$.
2. The area of the cross-section $A$. A wire that is twice as thick (in area) has half the resistance. $R \propto \dfrac{1}{A}$.
3. The material. A wire of copper and a wire of iron of the same size have different resistances.
4. The temperature. The resistance of a metal grows when the temperature rises.

The first two together give

The constant $\rho$ (the Greek letter rho) is the resistivity of the material. It does not depend on the shape of the wire. It tells how much a material resists the current. Its unit is the ohm metre ($\Omega\,\text{m}$).

When the unknown is $\rho$, we write $\rho = \dfrac{RA}{L}$.

Resistivity of some materials

• Metals have very small resistivity, and they are good conductors.
• Alloys (a mixture of metals) have a larger resistivity than the metals that they are made of, and their resistivity changes little with the temperature.
• Insulators have a huge resistivity. Almost no current flows through them.

You should find that a long, thin nichrome wire has a resistance of several ohms, while a thick copper wire of the same length has only a small fraction of an ohm.

Why these materials?

• Copper and aluminium are used for the wires that bring the current to our homes, because their resistivity is small, so little energy is lost as heat.
• Tungsten is used for the filament of a filament bulb. It has a very high melting point, so it can glow white hot without melting.
• Nichrome is used in heaters, toasters and irons. Its resistivity is large, and it does not burn easily in the air when it is red hot.

A worked example

A copper wire is $10\ \text{m}$ long and has an area of cross-section of $1.0\ \text{mm}^2$. What is its resistance?

Step 1. We change the area to square metres: $1.0\ \text{mm}^2 = 1.0 \times 10^{-6}\ \text{m}^2$.

Step 2. $R = \rho\dfrac{L}{A} = 1.68 \times 10^{-8}\ \Omega\,\text{m} \times \dfrac{10\ \text{m}}{1.0 \times 10^{-6}\ \text{m}^2} = 0.168\ \Omega$.

A $10\ \text{m}$ copper wire has a very small resistance. This is why connecting wires do not change a circuit.

Practise with new numbers

Try it in Python

The program finds the resistance of four wires of the same size but of different materials.

Remember these

• $R = \rho\dfrac{L}{A}$. The resistance is proportional to the length and inversely proportional to the area.
• Resistivity is a property of the material. The unit is the ohm metre.
• Metals have low resistivity, alloys higher, insulators very high.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A wire has a resistance of $4\ \Omega$. What is the resistance of a wire of the same material with twice the length and the same area?

2. The radius of a wire is doubled. What happens to its resistance?

3. Why are house wires made of copper and not iron?

4. A copper wire ($1.68 \times 10^{-8}\ \Omega\,\text{m}$) is $20\ \text{m}$ long, with an area of $0.5\ \text{mm}^2$. Find its resistance.