Resistivity (4.2.4)
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The resistance of a wire depends on the material of the wire and its physical dimensions, specifically its length and cross-sectional area.
Resistivity is a property of the material the wire is made of. Wires made of the same material may have different resistances due to their varying dimensions, but they still possess the same resistivity.
The equation for the resistance of a wire is:
Where:
- is the resistivity of the wire material in ,
- is the length of the wire, and
- is the cross-sectional area of the wire.
The resistivity of a metal can be determined using the following circuit components in the setup below:
- A flying lead is used to adjust the length of a current-carrying test wire being measured.
- A voltmeter is connected in parallel to the test wire using the flying lead to measure the potential difference across it.
- An ammeter is connected in series to the test wire to measure the current flowing through it.
- A power source supplies a voltage to the circuit.
The potential difference across the test wire and the current through it are measured, allowing the resistance to be calculated by Ohm’s law:
The equation for the resistance of a wire can be rearranged for resistivity :
A graph of resistance against length will have a gradient equal to the resistivity of the material divided by the cross-sectional area:
Which can then be multiplied by the cross-sectional area of the wire to find the resistivity.
The graph below shows the resistance against length plots for copper and aluminium wires of the same cross-sectional area.
The gradient for the aluminium wire is steeper than that of the copper wire, indicating that aluminium has a higher resistivity than copper.
The resistivity of a metal increases with temperature. When the temperature of a wire increases, the positive metal ions gain thermal energy and vibrate with a greater amplitude around their mean positions.
- Larger vibrations of the metal ions in a current-carrying wire result in more collisions between the electrons and the metal ions.
- A greater number of collisions leads to the electrons losing more energy as they move through the wire, causing them to move through the wire more slowly.
- Slower-moving electrons result in a lower current for a given potential difference and therefore a larger resistance.
In some semiconductors, an increase in temperature results in a decrease in resistance, unlike in metals. These semiconductors exhibit a negative temperature coefficient (NTC).
In NTC semiconductors, an increase in temperature leads to an increase in number density which is the number of charge carriers per unit volume.
A larger number density leads to an increase in current for a given potential difference, and hence a decrease in resistance.
Question walkthrough
Resistivity
Sketch how resistivity varies with temperature for an intrinsic semiconductor, linking the shape of the graph to how charge carrier number density changes with temperature.
A thermistor is an electrical component made from a negative temperature coefficient (NTC) semiconductor. The thermistor circuit symbol is a fixed resistor with a line through it.
A thermistor is a variable resistor because its resistance, and hence resistivity, decreases with temperature.
It is useful to note that thermistors are used in temperature-sensing circuits. Thermistors are used to monitor the temperature of complex electrical devices, such as computers and mobile phones, as well as household appliances like kettles and toasters.
The decrease in resistance of a thermistor with temperature is not linear.
Initially, the drop in resistance is quite steep, indicating a high sensitivity to small temperature changes. As the temperature continues to increase, the rate at which the resistance decreases becomes more gradual.
The resistance–temperature relationship for a thermistor can be determined using the experimental setup below.
- A waterproof thermistor is submerged in a hot water bath. The thermistor is connected to an ohmmeter to measure its resistance as the water cools.
- By taking readings of temperature and resistance using the thermometer and ohmmeter at set time intervals, respectively, you can observe the relationship between resistance and temperature of a thermistor experimentally.
Alternatively, if an ohmmeter is not available, the thermistor can be connected across a battery in series with an ammeter to measure the current, through it, and in parallel with a voltmeter to measure the potential difference, across it. The thermistor resistance can then be calculated from Ohm’s law:
The currents and voltages must be low to ensure the thermistor obeys Ohm’s law.