Physics A Level
Chapter 10: Resistance and resistivity 10.4 Resistivity
Physics A Level
Chapter 10: Resistance and resistivity 10.4 Resistivity
The resistance of a particular wire depends on its size and shape. A long wire has a greater resistance than a short one, provided it is of the same thickness and material. A thick wire has less resistance than a thin one. For a metal in the shape of a wire, R depends on the following factors:
- length L
- cross-sectional area A
- the material the wire is made from
- the temperature of the wire.
At a constant temperature, the resistance is directly proportional to the length of the wire and inversely proportional to its cross-sectional area:
$resistance\, \propto \,length$
and
$resistance\, \propto \,\frac{1}{{cross\, - \sec tional\,area}}$
We can see how these relate to the formulae for adding resistors in series and in parallel:
- If we double the length of a wire it is like connecting two identical resistors in series; their resistances add to give double the resistance. The resistance is proportional to the length.
- Doubling the cross-sectional area of a wire is like connecting two identical resistors in parallel; their combined resistance is halved (since $\frac{1}{{{R_{total}}}} = \frac{1}{R} + \frac{1}{R}$).
Hence the resistance is inversely proportional to the cross-sectional area.
Combining the two proportionalities for length and cross-sectional area, we get:
$resistance\, \propto \,\frac{1}{{cross\, - \sec tional\,area}}$
or
$R\, \propto \,\frac{L}{A}$
But the resistance of a wire also depends on the material it is made of. Copper is a better conductor than steel, steel is a better conductor than silicon, and so on. So if we are to determine the resistance R of a particular wire, we need to take into account its length, its cross-sectional area and the material. The relevant property of the material is its resistivity, for which the symbol is ρ (Greek letter rho).
The word equation for resistance is:
$\begin{array}{l}
resistance\, = \,\frac{{resistivity \times length}}{{cross - \sec tional\,area}}\\
R = \frac{{\rho L}}{A}
\end{array}$
We can rearrange this equation to give an equation for resistivity. The resistivity of a material is defined by the following word equation:
$\begin{array}{l}
resistivity = \frac{{resistance \times cross - \sec tional\,area}}{{length}}\\
\rho = \frac{{RA}}{L}
\end{array}$
Values of the resistivities of some typical materials are shown in Table 10.2. Notice that the units of resistivity are ohm metres ($\Omega m$); this is not the same as ohms per metre.
| Resistivity / $\Omega m$ | Material |
| $1.60 \times {10^{ - 8}}$ | silver |
| $1.69 \times {10^{ - 8}}$ | copper |
| $1.30 \times {10^{ - 8}}$ | $nichrom{e^{(a)}}$ |
| $3.21 \times {10^{ - 8}}$ | aluminium |
| $20.8 \times {10^{ - 8}}$ | lead |
| $44.0 \times {10^{ - 8}}$ | $mangani{n^{(b)}}$ |
| $49.0 \times {10^{ - 8}}$ | $eurek{a^{(c)}}$ |
| $69.0 \times {10^{ - 8}}$ | mercury |
| $800 \times {10^{ - 8}}$ | graphite |
| 0.65 | germanium |
| $2.3 \times {10^{ 3}}$ | silicon |
| ${10^{12}}$ | Pyrex glass |
| ${10^{13}} - {10^{16}}$ | $PTF{E^{(d)}}$ |
| $5 \times {10^{ 16}}$ | quartz |
(a) Nichrome – an alloy of nickel, copper and aluminium used in electric heaters because it does not oxidise at ${1000^ \circ }C$.
(b) Manganin – an alloy of $84\% $ copper, $12\% $ manganese and $4\% $ nickel.
(c) Eureka (constantan) – an alloy of $60\% $ copper and $40\% $ nickel.
(d) PTFE – Poly(tetrafluoroethene) or Teflon.
Resistivity and temperature
Resistivity, like resistance, depends on temperature. For a metal, resistivity increases with temperature.
As we saw earlier, this is because there are more frequent collisions between the conduction electrons and the vibrating ions of the metal.
Questions
8) Use the resistivity value quoted in Table 10.2 to calculate the lengths of $0.50 mm$ diameter manganin wire needed to make resistance coils with resistances of:
a: $1.0\,\Omega $
b: $5.0\,\Omega $
c: $10\,\Omega $.
9) $1.0\,c{m^3}$ of copper is drawn out into the form of a long wire of cross-sectional area $4.0 \times {10^{ - 7}}\,{m^2}$.
Calculate its resistance. (Use the resistivity value for copper from Table 10.2.)
10) A $1.0 m$ length of copper wire has a resistance of $0.50\,\Omega $.
a: Calculate the resistance of a $5.0 m$ length of the same wire.
b: What will be the resistance of a $1.0 m$ length of copper wire having half the diameter of the original wire?
11) A piece of steel wire has a resistance of $10\,\Omega $. It is stretched to twice its original length. Compare its new resistance with its original resistance.
EXAM-STYLE QUESTIONS
1) An element of an electric fire is made up from a length of nichrome wire of diameter $0.40 mm$ and length $5.0 m$.
The resistance of this element is ${R_1}$.
Another element, also made from nichrome, for a different electric fire, has a length of $2.0 m$ and a diameter of $0.20 mm$. This element has a resistance of ${R_2}$.
What is the relationship between ${R_1}$ and ${R_2}$? [1]
A: ${R_2} = 0.80\,{R_1}$
B: ${R_2} = 1.6\,{R_1}$
C: ${R_2} = 5.0\,{R_1}$
D: ${R_2} = 10\,{R_1}$
2) This is a circuit.
Which line in the table shows the changes to the lamp and the voltmeter reading when the temperature rises? [1]
| Lamp | Voltmeter reading | |
| A | gets brighter | decreases |
| B | gets brighter | increases |
| C | gets dimmer | decreases |
| D | gets dimmer | increases |
3) This shows the I–V characteristic of an electrical component.
a: Calculate the resistance of the component when the potential difference across it is:
i- $2.0 V$ [2]
ii- $5.0 V$. [1]
b: Suggest what the component is. [1]
[Total: 4]
4) A student connects a thermistor to a battery and an ammeter. He places the thermistor in a beaker of water and gradually heats the water from ${10^ \circ }C$ to its boiling point, recording the value of the current as he does so. He then plots a graph of the current in the thermistor against the temperature of the water.
a: Sketch the graph you would expect the student to obtain from the experiment. [1]
b: Explain how the student could now use the thermistor as a thermometer. [2]
[Total: 3]
5) a: Describe the difference between the conduction processes in copper and in silicon, a semiconductor. [3]
b: Explain why the resistance of a metallic conductor increases with temperature while that of a semiconductor decreases. [3]
[Total: 6]
6) A nichrome wire has a length of $1.5 m$ and a cross-sectional area of $0.0080\,m{m^2}$. The resistivity of nichrome is $1.30 \times {10^{ - 8}}\,\Omega m$.
a: Calculate the resistance of the wire. [2]
b: Calculate the length of this wire that would be needed to make an element of an electric heater of resistance $30\,\Omega $. [2]
[Total: 4]
7) This is a circuit.
a: When switch S is open the current in ammeter A is $0.48 A$. Calculate the e.m.f. of the battery. You may assume the battery has negligible internal resistance. [2]
b: When switch S is closed the current in the ammeter increases to $0.72 A$.
i- Determine the current in the $6.4\,\Omega $ resistor. [1]
ii- State the current in the thermistor. [1]
c: State and explain how the reading on the ammeter changes when the temperature of the thermistor is increased. [3]
[Total: 7]
8) a: Explain why the resistance of a metal increases when its temperature increases. [2]
b: State two other factors that determine the resistance of a stated length of wire. [2]
c: When a potential difference of $1.5 V$ is applied across a $5.0 m$ length of insulated copper wire, a current of $0.24 A$ is measured in it.
i- Calculate the resistance of the length of wire. [2]
ii- The resistivity of copper is $1.69 \times {10^{ - 8}}\,\Omega \,m$. Calculate the diameter of the wire. [3]
d: The wire is now made into a tight bundle. State and explain how you would expect the current in it to change. [3]
[Total: 12]
9) This diagram shows a piece of silicon of width $32 mm$ and length $36 mm$. The resistance of the silicon between the points P and Q is $1.1\,M\Omega $. Silicon has a resistivity of $2.3 \times {10^3}\,\Omega \,m$.
a: Calculate the thickness of the piece of silicon. [3]
b: Calculate the current that would pass through the silicon if a potential difference of $12 V$ were applied across P and Q. [2]
c: Describe how the current would change if it were large enough to cause the silicon to become significantly warmer. [3]
[Total: 8]
10) A student is investigating the properties of a semiconducting diode. This diagram shows the circuit she builds.
a: i- Sketch a graph to show how the current in the diode would vary as the voltage across it is increased from $0 V$ to $1.0 V$. [1]
ii- The supply is now connected in the reverse direction and once more the potential difference across the diode is increased from $0 V$ to $1.0 V$.
Complete the I–V graph. [1]
b: Suggest why the safety resistor is required. [2]
c: When the potential difference across the safety resistor is $1.4 V$, the current in it is $20 mA$. Calculate the resistance of the safety resistor. [2]
[Total: 6]
11) a: Explain what is meant by an ohmic conductor. [2]
b: i- Sketch a graph of resistance R against voltage V for a wire of pure iron kept at constant temperature. Label this line X. [1]
ii- Sketch a graph of resistance R against voltage V for a second wire of impure iron, of the same diameter and the same length, which is kept at the same temperature. Label this line Y. [1]
iii- Explain how the graphs would change if the wires were kept at a higher, but still constant, temperature. [1]
c: Deduce how the resistance of a wire made of pure iron would change if both the diameter and the length were doubled. [3]
[Total: 8]
12) The readings in this table are recorded from an experiment to measure the resistivity of silver.
| Diameter of the wire | $0.40 \pm 0.02\,mm$ |
| Length of the wire | $2.25 \pm 0.05\,m$ |
| Resistance of the wire | $0.28 \pm 0.01\,\Omega $ |
a: Calculate the resistivity of silver. [2]
b: i- Calculate the percentage uncertainty in each of the variables. [2]
ii- Use your answers to i to calculate the absolute uncertainty in the value of the resistivity obtained in the experiment. [2]
[Total: 6]
SELF-EVALUATION CHECKLIST
After studying the chapter, complete a table like this:
| I can | See topic… | Needs more work | Almost there | Ready to move on |
| state and understand Ohm’s law | 10.2 | |||
| recognise ohmic and non-ohmic components | 10.2, 10.3 | |||
| recognise and understand the changes in the resistance of metals and thermistors when there is a change in their temperature | 10.3, 10.4 | |||
| understand that a light-dependent resistor is a component whose resistance decreases as the light level increases | 10.3 | |||
|
understand that resistivity $\rho $ of a material is defined as: $\rho = \frac{{RA}}{L}$ where R is the resistance of a wire of that material, A is its cross-sectional area and L is its length. The unit of resistivity is the ohm metre ($\Omega \,m$). |
10.4 |