A conductor with a larger surface area has a greater number of conduction electrons, as it has a larger volume. For a given length, a conductor with a larger cross section area has a greater number of electrons carrying the current. Hence each electron experiences fewer collisions and therefore the resistance of the conductor is smaller.
Different materials have different electron densities (number of electrons per unit volume). Therefore the resistance of two different conductors (same length and cross sectional area) will be different because the current will be carried by different amounts of electrons.
As the temperature is increased, the atoms in the conductor gain thermal energy and vibrate more. This results in more collisions between the conductor’s atoms and the conduction electrons. Hence the resistance increases
The relationship between the resistance, R, length l and cross sectional area A is:
Where ρ is the resistivity of the conductor, which is measured in ohm metres (Ωm). The resistivity of a conductor depends on the material. This equation shows that the resistance of a conductor is directly proportional to its length and inversely proportional to its cross sectional area. It can also be seen that if a graph of resistance (R) versus length (l) is drawn, the gradient will be given by ρ /A. Therefore if the cross sectional area of the wire is known, the resistivity of the wire can be determined from the gradient.
Using ohms law, the resistance of a conductor can be determined by measuring the potential difference across it and the current flowing through it. In this experiment I will investigate the effect of length on the resistance of a conducting wire.
Aim: To investigate the effect of length on the resistance of a conducting wire.
The length of a conducting wire will be varied, and the resistance determined by measuring the potential difference and the current and then using ohms law. The following variables will be kept constant in order to ensure a fair test: temperature, cross sectional area and type of wire.
Prediction: If the length of the wire is doubled, the resistance of the wire will also double (i.e. the resistance is directly proportional to the length). This is because as the length of a conductor is increased, there are more collisions between the conduction electrons and the conductor’s atoms. Therefore it is more difficult for electrons to go all the way through the conductor and hence the resistance is larger. If the conductor is twice as large there will be twice as many collisions and hence the resistance of the conductor will be doubled. A graph of resistance versus length should look like:
METHOD
List of Apparatus:
- Copper wire (0.9mm thickness)
- Amp meter
- Voltmeter
- Power supply
- Meter rule
- Crocodile clips
The following circuit was setup to determine the resistance of the varying lengths of wire:
- The experiment was setup as shown in the diagram above. The ammeter is connected in series and the voltmeter in parallel.
- The crocodile clip was placed at the 25 cm mark, and the reading on the voltmeter and ammeter was noted. It is important to take the measurements quickly as resistive heating will affect the resistance of the wire (as the wire heats up, its resistance will increase). When taking the readings, the value on the voltmeter and ammeter was fluctuating slightly, so we took the approximate average values.
- This was repeated for 10 cm length intervals up to 95cm. In between the measurement we allowed some time for the wire to cool down so that the experiment was fair.
- The resistance of the different lengths of wire was calculated using ohms law.
- The experiment was then repeated three times and the results where averaged, in order to get more precise results. A graph of the results was then plotted.
DISCUSSION
The results obtained are summarised in Table1 and graph 1.
The graph obtained shows very good agreement with the predicted graph (i.e. a straight line through the origin). The gradient of the graph is given by ρ/A. We find the gradient to be 2.68 Ωm-1. To find the resistivity of the material we have to multiply this by the cross sectional area of the wire.
The diameter, d, of the wire is 0.09mm (measured using a micrometer) .
➔ Therefore the radius r =
➔ r = 0.045 mm = 4.5 × 10-5 mm
The cross sectional area of the wire is given by πr2 .
➔ π(4.5 × 10-5) 2 = 6.36 × 10-9 m2
➔ Resistivity, ρ = 2.68 × 6.36 × 10-9 = 1.7 × 10-8
This compares very well with the actual value of 1.8×10-8 Ωm for copper. As our experimental value for the resistivity is consistent with the actual value to within experimental errors, we can conclude that our experimental results are very accurate.
I also predicted that if the length of the wire were doubled, the resistance of the wire would also double. The table below shows the resistance of different lengths of wire, taken from graph 1.
It can be see from the table that as the length of wire is doubled the resistance also approximately doubles. This is consistent with the quantitative prediction to within experimental errors. Overall the results show very good agreement with my prediction.
CONCLUSION
Overall the results showed excellent agreement with the prediction. This is partly due to the relatively simple experimental setup. I predicted that the resistance would be directly proportional to the length of the wire and resulting graph would be a straight line thought the origin. The results obtained where consistent with this. I also predicted that if the length of the wire were doubled, the resistance would also double. This was also observed from the results obtained. The experiment was repeated three times, and the results averaged. There was good agreement between the different sets of results, and throughout the experiment we did not record an anomalous result. From the gradient of the graph of resistance versus length (graph 1) we obtained the resistivity of the copper wire. The value obtained was consistent with the actual value, which further verifies the accuracy of our results.
The errors in the experiment where due to the resistive heating of the wire and the fluctuations in the output of the ammeter and voltmeter. Resistive heating increases the resistance of the wire, and this can happen very quickly if a large current flows. The error due to heating was minimised by taking the V-I readings as quickly as possible, before the wire had time to heat up significantly. Also we allowed time in between taking measurements for the wire to cool down. To reduce the error due to the fluctuations in the ammeter and voltmeter, we tried to determine the mean value of the fluctuating reading. Also we repeated the experiment three times in order to reduce any random errors due the fluctuations and any other human errors.
The experiment could be extended by investigating whether the relationship between length and resistance is maintained when wires of different materials and different cross sections are used. It could be further extended by investigating other variables, which affect the resistance of a wire (i.e. cross sectional area, type of material and temperature).
