Measuring the Resistivity of Copper Wire of Different Lengths

In this report, I will be writing about the experiment I will conduct on copper wire of different lengths. The dependent variable I will be measuring is the resistance of the Copper wire. To do this experiment, one needs to obtain measurements with a high degree of accuracy, taking care of the equipment they use, and measuring each value to a certain degree of accuracy for all results. The problem with measuring the resistivity of Copper wire is due to the properties of copper as a material.

Copper naturally has a low resistance due to it being a superconductor, meaning that it only has a resistance of minute amounts. As it has this property, it is important to use a copper wire specimen that is long enough and thin enough to have an appreciable resistance. The normal value for the resistivity of copper is about 10-8Ωm. A 1m length of copper wire with a cross-sectional area of 1mm² (10-6m²) can be predicted to have a resistance of 0.01Ω. This can be calculated by using the resistance formula of: R=ρlA≈ 10-8 Ωm x 1m10-6m2=10-2Ω

The wire I will use is going to be thinner than this and will vary in length from 0. -1. 0 metres with a difference of 0. 2m from the previous wire specimen. In total, I will have 5 different lengths. Apparatus:

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• Voltmeter- Accuracy stated as (± 0. 5% Read. + 1dgt) in the user manual
• Ammeter- Accuracy stated as (± 1. 2% Read. + 1dgt) in the user manual
• Battery Supply of 6V
• Copper Wire
• 1m Ruler in cm
• Scissors
• Electrical Wires
• Crocodile clips
• Micrometer

Method: The following procedure described below is how I intend to gain my results:

1. I will measure out the different lengths of copper wire I intend to use using a millimeter ruler to gain the most accurate results I can.

2. Once the lengths are cut, the diameter of the copper wire I am using must be measured. To gain the most accurate result, I will use a micrometer and measure the diameter in several places on the wire and take an average value from these readings to work out the average cross-sectional area.

3. I will connect the first length of wire into an electrical circuit, making sure that current can flow through the entire length of the copper wire connected. The circuit will look like this diagram:

V V A A

4. The voltage will be recorded across the wire and the current running through it.

5. To find the resistance of the wire I will use the formula V=IR.

6. The resistivity can then be worked out using the formula: ρ =RAL where R is the resistance calculated, A is the cross-sectional area of the copper wire calculated and L is the length of the copper wire. The measurements shall be recorded in the following table shown below:

Resistivity of Wire

The physical properties of a wire can either be categorized as being an intrinsic property of an extrinsic property. The difference between the two categories of properties is that intrinsic properties do not depend on the amount of material that is present, whereas extrinsic properties do depend on the amount of material that is present.

In the following investigation of the resistivity of copper wire, one could say that the value of the voltage, resistance and current are all intrinsic properties of the copper wire. The extrinsic value of the copper wire would be its resistivity. The resistivity of the copper wire will be dependent on the material itself, which is copper. The resistivity of a material can be defined as the resistance of a 1m length with a 1m² cross-sectional area. As the resistivity of material depends mainly on the properties of the material itself, each material whether it is copper or pure silicon has its own resistivity coefficient. The coefficient for copper is 1.72 × 10-8Ωm. This value may seem very small for resistivity, but if one were to know that copper is classed as a superconductor meaning that it conducts electricity extremely well, they would know that in order for the conductance to be very high, the resistivity must be very low. This can also be explained by the fact that resistivity is the inverse of conductivity (σ =1/ρ ). The potential difference across the copper wire (measured in volts) and the flow of charge (the current) through the copper wire are related through the resistance of the copper wire, not its resistivity.

In order to find the resistivity, one needs to work out the resistance first by using the equation R=VI, and then from this they can use the formula ρ =RAL to find the resistance. The “A” represents the cross-sectional area of the wire that will be used in the experiment. The resistance of the wire is expected to double in value when the length of the wire doubles in size. The resistivity, however, should stay near enough the same throughout all of the repeats conducted. Reducing the uncertainty in the results

There are some factors that could affect the accuracy of my results in the experiment of the resistivity of copper wire. One of the factors which could affect the accuracy of my results is to do with the measuring devices I use to conduct the experiment. Any measuring device can only be used to measure a certain degree of accuracy. It is this certain degree that determines how accurate your results are to the true value. In my experiment, I am using a 3½ Digits Multifunction Multimeter (DMM) to measure the current through the circuit and the potential difference (p. d. ) across the copper wire.

The main advantage of using a DMM compared to using an analog voltmeter is the fact that they allow you to record value to a certain number of decimal places by having different ranges that correspond to the level of precision of the reading. In the experiment I am conducting, I will be measuring the p. d. to a resolution of 0. 001V using the 2V range on the multimeter. Having the resolution to this degree of measurement ensures that I get a voltage reading to 3 decimal places increasing the accuracy of the reading and allowing me to obtain a closer value to the true value.

The accuracy of the ammeter has been published as being ±1. 2% of the reading + 1 LSD for the range (200mA) and resolution (0. 1mA) I will be using for the current. This means that the value I will record will be 1. 2% of the true value of the current +0. 1mA. I am using the 200mA range rather than the 20A range because the resolution of the result is greater than that of the 20A range. This will record a more accurate result which reduces the uncertainty in my results. Similarly, the range I will use on the voltmeter which is at 2V has an accuracy of ±0. 5% of the reading + 1 LSD, which is even more accurate.

Another factor that can affect the resistivity of the result is the temperature of the copper wire. This can affect the resistivity by changing the value of the resistance to make the resistance less proportional to that of the length of wire. Normally the resistance of the wire will increase as the length of the wire increases due to their being more atoms in the wire for the electrons to pass by in order to get through the entire length of wire. As the increase in resistance α increase in length, the resistance should double when the length of the copper wire is doubled.

In order to try and make sure the resistance is not affected by temperature, I will connect the copper wire up into the circuit at a low voltage so that the copper wire will not warm up and increase in resistance due to the atoms inside vibrating more. I will also be using a micrometer to measure the diameter of the wire. I am using a micrometer instead of a standard cm ruler because the level of uncertainty is far less than that of a ruler. The micrometer allows me to record a value for the diameter of the wire with an uncertainty of ±0. 0005mm, whereas with an ordinary ruler with mm markings, the uncertainty would be ±0. 1mm.

Results:

These are the results I collected from the experiment carried out. All of the data is raw data that I have collected myself and has not been manipulated in the way at all. N. B- The diameter of the wire was measured to be 0. 435mm. The cross-sectional area was calculated as being 1. 48 × 10-7m2. This value was used throughout the experiment to work out the different resistivity values using the resistivity equation as stated previously.

From the table above, I also worked out the averages of the results measured from the experiment.

Uncertainties within my results: Before creating the graph of my results, I calculated the overall uncertainties of each measurement within this experiment, so that I could see where the most uncertainty of the average resistivity value comes from.

To calculate the uncertainty for each measurement, I took the average measurement that had the biggest difference from its original data. The Percentage of uncertainties of each measurement was as follows:

• Percentage uncertainty of the Voltage V= 0. 206±0. 005 V Uncertainty in V= 0. 005/0. 206 × 100% ≈ ±2. 43%
• Percentage uncertainty of the Current I=1. 729±0. 019 A Uncertainty in I=0. 019/1. 729 × 100% ≈ ±1. 10%
• Percentage of Uncertainty in Resistance R=V/I Uncertainty of R=1. 10%+2. 43% ≈ ±3. 53%
• Percentage of Uncertainty in Length Uncertainty=0. 6±0. 001m

Uncertainty in L=0. 001/0. 6 × 100% ≈ ±0. 17%

Percentage of Uncertainty in Area: The Diameter of the wire is 0. 435±0. 0005mm

The best area where the diameter is 0. 435mm A=? 0. 21752? 0. 1486mm2? 1. 486 × 10-7m2

The Maximum area where the diameter is ? 0. 4355mm A=? 0. 217752? 0. 1489mm2? 1. 489 × 10-7m2

The Minimum area where the diameter is ? 0. 4345mm A=? 0. 217252? 0. 1482mm2? 1. 482 × 10-7m2

So the area is 0. 148±0. 0004mm2 with a percentage uncertainty of: A=0. 0004/0. 148 × 100% ≈ ±0. 27%

• So the percentage uncertainty in the Resistivity can be calculated as the sum of all the uncertainties in the experiment:

ρ =RAL=3. 53%+0. 27%+0. 17%=±3. 97%

The percentages of instrument error are as follows:

• Voltmeter reading is ±0. 0005V Instrumental error in Voltmeter= 0. 0005/0. 206 × 100 ≈ 0. 24%
• Ammeter reading is ±0. 0005A Instrumental error in Ammeter=0. 0005/1. 729 × 100 ≈ 0. 03%
• Micrometer reading is ±0. 0005mm Instrumental error in Mircometer=0. 0005/0. 435 × 100 ≈ 0. 11%
• The total instrumental error is the total of each instrumental error stated above which would be 0. 38%.

Graph 1: Graph 2:

Data Analysis: In all of the results that I have collected, there is a strong relationship between the increasing length of wire and the value for the resistance.

One would expect this strong correlation between the resistance and the length since one of the simple laws of electrical resistance is that it increases proportionally with the increase in the length of the wire. One can explain this through the understanding of electrons in a circuit and the atoms arranged within the components in a circuit. With my experiment of copper wire, a current passed through my circuit once a voltage was applied to the circuit. When the electrons were given the energy to move they passed through the circuit to the copper wire where they experienced the resistance which was calculated.

As the lengths of the copper wire increase, the number of fixed atoms within the structure of the wire increases. Due to this, the electrons have a higher chance of colliding with the fixed atoms, which causes the wire to heat up and increase the resistance. One can see the certainty in the correlation between the average resistance and the length of the copper wire by looking at the gradient of the line of best fit within graph 1. The gradient shows that R? =0. 9984, showing an extremely strong positive correlation between the two variables.

From the equation of the gradient displayed in graph 1, the average resistivity can be calculated which takes into account all of the points within the data collected. The gradient of the line shows the equation Resistance (R)Length (L). In the calculation for resistivity, one not only needs the value of RL but also needs the cross-sectional area of the wire. If the cross-sectional area of the wire is multiplied by the gradient, then the average resistivity can be calculated: ρ =RAL=0. 1192? 1. 486 × 10-7m2? 1. 77 × 10-8Ω m

In Graph 2, the percentage of the uncertainty of each average resistance was displayed in the vertical error bars.

The percentage of the uncertainty of the length of the wire was so small that it was not worth adding to the graph since it is extremely hard to see on the graph. From these percentage uncertainties of the average resistance in the experiment, one can calculate the maximum and the minimum values for the resistivity from looking at the gradients like we did for graph 1. To calculate the minimum gradient, I took the gradient of the line from the maximum uncertainty in the lowest resistance to the minimum uncertainty of the highest resistance.

I did this to obtain the shallowest gradient possible from all the points on the graph. I then multiplied this gradient by the smallest area value. lowest ? =0. 1144? 1. 482 × 10-7m2? 1. 70 × 10-8Ω m For the maximum value of resistivity, I took the value of the gradient of the line from the minimum uncertainty in the lowest resistance to the maximum uncertainty of the highest resistance. I did this to obtain the steepest gradient possible from all of the points on the graph. I then multiplied this by the maximum area.

maximum ρ =0. 1263? (1. 489 × 10-7m2)? 1. 88 × 10-8Ω m

After looking at the average, minimum and maximum values of the resistivity taking into account all of the uncertainties within the calculation one could say that from the investigation conducted, the resistivity of copper wire is 1. 76 × 10-8±1. 2 × 10-9. The percentage uncertainty of the resistivity would then be: 1. 2 × 10-9/1. 76 × 10-8 × 100% ≈ 6. 8% Biggest Source of Uncertainty From looking at all of the percentage uncertainties for all my measurements, the resistance produced the most uncertainty. The uncertainty of resistance was worked out by adding up the uncertainty of the voltage and the current measured.

It must have been from these two calculations where the uncertainty of the resistance became noticed. From calculating the instrumental errors of the multimeter used as a voltmeter and an ammeter, I would not conclude that the vast majority of the error came from the accuracy of the apparatus. I would say that the average resistance I calculated was from the average current which had the biggest difference from its original data, and the average voltage which had the biggest difference from its original data. The average data I had chosen was 0. 206±0. 05V and the average data I had chosen for the current was 1. 729±0. 019A, as they had the biggest uncertainties. Due to this fact, I would have produced an uncertainty that had the biggest difference from the original value, so the maximum possible uncertainty for the resistance. Anomalies and Systematic Errors I did not have any anomalous results when looking at the average resistance graph. All of the points plotted to show a strong correlation with the increase in length. Systematic errors may have contributed to some of my resistivity values being higher or lower than my overall average.

An example of this could have been when measuring the diameter of the copper wire. The micrometer did not let me know if both of the sides of the copper wire were touching the micrometer measuring device sufficiently enough or whether or not it was touching both sides of the copper wire more than enough, which would then mean it squashed the diameter of the wire resulting in a lower diameter at certain points across the wire since I took 3 readings and averaged them out. If this was the case, then one of my wires may have had a higher resistance than the others.

One other systematic error may have come from the battery pack. It may have had a temporary glitch in which less electrical energy was sent through the circuit meaning less current was flowing through the circuit, resulting in a larger resistance than that of the previous recording with the same length of wire. This would also alter the final value of the resistivity. Another uncertainty that would be counted as the human error could have been the position at which I had placed the crocodile clips at either end of the copper wire.

For the same length of wire, the crocodile clip may have been placed further away from the end of the copper wire than the previous measurement, meaning that the length of the wire would have decreased marginally which may have resulted in a lower resistance recording. Also, when I measured the length of the copper wire, I had to straighten out the length of the wire since it was coiled. When doing this I may have accidentally pulled the length of the wire increasing its length by a fractional amount.

Having said this, it may have altered the resistance measured in the wire making it larger than it should have been since the electrons have to travel a longer distance. Evaluation After looking at all of my results, I believe that the method I used and the ways of reducing the uncertainty in my experiment were effective. The instrumental errors were minimal and the overall uncertainty of my final calculation of resistivity was a low value. The resistivity value itself did alter but mainly stayed constant throughout the experiment.

As I have said, I do not believe this was because of the accuracy of the multimeters I used but due to other factors such as changes in the environment like temperature, or due to systematic errors to do with the battery pack I used. To decrease the uncertainty in my resistance measured, I could use an even lower resolution on my voltmeter (0. 1mV) and ammeter (0. 001mA) to reduce the negative effect of Least Significant Digits (LSD) and to give the most accurate result.

This way I could then increase the precision of my results and record a value which is closer to the true value When comparing my average value of resistivity with the published value of resistivity which is 1. 72 × 10-8Ω m, my average value is very close to the published value which shows the level of accuracy throughout my experiment considering the more precise tools that were used by the professionals to gain the published value. The repeats I did help me to record a value for the resistivity that was close to the published value by reducing the random uncertainty in my results.

To gain even more accuracy I could do more repeats, or I could alter the intervals between each length to 0. 1m to increase my range of data. That way I will reduce even more random error within my data. I could also change the different diameters of the wire or change the material I use to compare these results with those and see how they differ. One other change I could do next time is to use an Alternating Current (AC) rather than a Direct Current (DC) since AC is more conventional in houses so it would have provided further information as to how good copper is in the use of houses.

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