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Resistance of wires

Electricity and magnetism · level 5, lesson 5 of 6

A question to start

Power cables that carry electricity across a country are thick, and the cable to an electric stove is thicker than the one to a lamp. Why does thickness matter?

The idea

The resistance of a wire depends on its length \(L\), its cross-section area \(A\) and what it is made of:

\[R = \frac{\rho L}{A},\]

where \(\rho\) (rho) is the resistivity of the material, in ohm metres (\(\Omega\) m).

Material Resistivity at 20 °C
Silver \(1.59 \times 10^{-8}\ \Omega\) m
Copper \(1.68 \times 10^{-8}\ \Omega\) m
Aluminium \(2.65 \times 10^{-8}\ \Omega\) m
Nichrome (heater wire) about \(1.1 \times 10^{-6}\ \Omega\) m

Double the length, double the resistance. Double the area, half the resistance.

A wire carrying current \(I\) turns energy into heat at a rate \(P = I^2R\). So a thick wire, with a small \(R\), wastes less and stays cooler.

LAresistivity ρ

Worked examples

1. A house wire. 10 m of copper with \(A = 1.5\) mm² \(= 1.5 \times 10^{-6}\) m²: \(R = 1.68\times10^{-8} \times 10/1.5\times10^{-6} = 0.11\ \Omega\).

2. Heat in the wire. At 15 A: \(P = 15^2 \times 0.11 = 25\) W, spread along 10 m.

3. Nichrome. 1 m of nichrome with \(A = 0.1\) mm²: \(R = 1.1\times10^{-6} \times 1/10^{-7} = 11\ \Omega\). Good for a heater, bad for a cable.

Try it

  1. Find the resistance of 100 m of copper wire with \(A = 2.5\) mm².
  2. An aluminium wire has the same size. Is its resistance larger or smaller?
  3. A wire is cut in half. What happens to the resistance of each half?
  4. Why does a thin extension cord get warm when it runs a heater?
Answers
  1. \(1.68\times10^{-8} \times 100/2.5\times10^{-6} = 0.67\ \Omega\).
  2. Larger, by \(2.65/1.68 = 1.6\) times: \(1.06\ \Omega\).
  3. It halves.
  4. Its resistance is high, and a heater draws a big current, so \(I^2R\) heats the cord.

Common mistake

Forgetting to change mm² to m². 1 mm² is \(10^{-6}\) m², not \(10^{-3}\).

Where it leads