Temperature and the resistor colour code

A bulb that fails when it is switched on

Old filament bulbs usually fail at the moment they are switched on, not while they glow. The reason is that a cold filament has a much smaller resistance than a hot one. At the first moment the current is large. Let us see why resistance changes with temperature.

Metals: resistance rises with temperature

In a metal the ions vibrate more as the metal gets hotter. The electrons hit them more often, so the relaxation time $\tau$ gets smaller. The number $n$ hardly changes. Since $\rho = m/(ne^2\tau)$, the resistivity goes up.

Over a modest range of temperature, the rise is nearly in proportion to the temperature:

Here $\rho_0$ is the resistivity at a reference temperature $T_0$. We take $T_0 = 20\ ^\circ$C. The number $\alpha$ is the temperature coefficient of resistivity, in per $^\circ$C. The resistance of a wire changes in the same way:

For copper $\alpha = 3.9 \times 10^{-3}$ per $^\circ$C. For silver it is $3.8 \times 10^{-3}$, for tungsten $4.5 \times 10^{-3}$ and for iron $5.0 \times 10^{-3}$.

A worked example. A $100$ W lamp for $230$ V has a tungsten filament that glows at $2500\ ^\circ$C. Find its resistance when hot and when cold, and the current at the moment of switching on.

1. Hot resistance. The lamp draws $100$ W at $230$ V, so $R_{\text{hot}} = \dfrac{V^2}{P} = \dfrac{(230\ \text{V})^2}{100\ \text{W}} = 529\ \Omega$.
2. The factor. $1 + \alpha(T - T_0) = 1 + 4.5 \times 10^{-3} \times (2500 - 20) = 12.2$.
3. Cold resistance. $R_{\text{cold}} = \dfrac{529\ \Omega}{12.2} = 43.5\ \Omega$.
4. Switching on. $I = \dfrac{230\ \text{V}}{43.5\ \Omega} = 5.3$ A. The lamp draws only $\dfrac{100\ \text{W}}{230\ \text{V}} = 0.43$ A once it glows.

The formula is only roughly right over such a wide range. It still shows the size of the effect: the first current is about twelve times the steady one.

Set $\alpha$ to a negative value and raise the temperature. What does the resistance do?

Alloys, semiconductors and superconductors

• Alloys. Manganin and constantan have an almost zero $\alpha$ ($2 \times 10^{-5}$ per $^\circ$C for manganin). Their resistance hardly changes, so they are used in standard resistors and resistance boxes. Nichrome has a high resistivity, a small $\alpha$ and does not oxidise when red hot. It is used for heating elements.
• Semiconductors. Carbon, silicon and germanium have a negative $\alpha$. A rise in temperature frees many more charge carriers. The gain in $n$ beats the loss in $\tau$, so the resistivity falls. A thermistor is a semiconductor resistor that is used to measure temperature.
• Superconductors. Below a critical temperature, the resistance of some materials drops to zero. Mercury does so at $4.2$ K. A current started in a superconducting ring goes on flowing with no battery.

Devices that do not follow Ohm's law

Ohm's law, $V \propto I$, holds for ohmic materials at a steady temperature. It is not a law of nature for every device. Look at three graphs of current against voltage.

• A metal resistor gives a straight line. The resistance is the same at every voltage.
• A filament lamp gives a curve. More current heats the filament, its resistance rises, and the slope falls.
• A diode passes current mostly in one direction, and only after about $0.7$ V. Its graph is far from a straight line.

The colour code

A carbon resistor is small, so its value is printed as coloured bands. Hold it with the bands at the left. The first two bands are digits. The third is the multiplier, a power of ten. The last band is the tolerance.

The colours stand for the digits black $0$, brown $1$, red $2$, orange $3$, yellow $4$, green $5$, blue $6$, violet $7$, grey $8$ and white $9$. A rhyme that helps: B B ROY of Great Britain has a Very Good Wife. A gold band means $\pm 5\ \%$, silver $\pm 10\ \%$, and no fourth band $\pm 20\ \%$.

The resistor in the picture is yellow, violet, orange, silver. That is $47 \times 10^3\ \Omega = 47\ \text{k}\Omega$, with a tolerance of $\pm 10\ \%$. Its true value lies between $42.3\ \text{k}\Omega$ and $51.7\ \text{k}\Omega$.

Practise with new numbers

Try it in Python

The program turns the colours of three resistors into a value and a range.

Common mistakes

• Using $T$ in place of $T - T_0$. The change in resistance depends on the rise in temperature above the reference.
• Thinking every conductor has a positive $\alpha$. Semiconductors have a negative one.
• Trusting the linear formula over a huge range. It is a good rule for a few hundred degrees at most.
• Thinking Ohm's law holds for every device. A lamp and a diode do not obey it.
• Reading the multiplier band as a plain digit. Orange as a multiplier is $10^3$, not $3$.
• Reading the bands from the wrong end. The gold or silver band comes last.

Remember these

• Metals: $R_T = R_0[1 + \alpha(T - T_0)]$, with $\alpha$ positive. Resistance rises with temperature.
• Semiconductors: $\alpha$ is negative. Alloys such as manganin: $\alpha$ is almost zero.
• Superconductors: the resistance is zero below a critical temperature.
• Ohm's law holds for ohmic materials only, at a steady temperature.
• Colour code: digit, digit, power of ten, tolerance. Gold $\pm 5\ \%$, silver $\pm 10\ \%$, none $\pm 20\ \%$.

Check yourself

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

1. Why is the resistance of a glowing filament so much more than its resistance when cold?

2. Why is manganin used for standard resistors?

3. A thermistor is joined to a fixed battery. As the thermistor warms, its resistance falls. What happens to the current?

4. A resistor has the bands brown, black, red, gold. What are its value and its tolerance?