Resistivity

Aim:To investigate the physical and electrical properties of a wire in order to find the resistance using Ohms law and from this to find the Resistivity of the wire at room temperature.


Hypothesis:
Electricity is the flow of charged particles around a complete circuit; a circuit consists of three main factors, voltage, current and resistance.

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Ohms Law states that at a constant room temperature these factors can be related by the equation,
V = IR
Where, V = Voltage (V)
I = Current (A)
R = Resistance (?)
Therefore, we can say that at a constant room temperature resistance is relative to the value of the voltage and of the current, this can be expressed by the equation,
By altering the values of voltage and current the value of resistance can be altered at any given point in a circuit. In addition to the relationship stated in Ohms Law, resistance is related to length and cross sectional area.


By saying that resistance is due to collisions between electrons we can deduce that resistance is proportional to length because by doubling the length of a wire we double the chance of a charged particle colliding (resistance), therefore,
R ? L
Also, if we double the area it will consequently double the number of charged particles flowing at a given point and thus double the current and halving the resistance. Therefore,
By combining these equations we can produce,
Where, = Resistivity (?/m) (material constant)
R = Resistance (?)
a = Cross Sectional Area (mm)
L = Length (m)
Resistivity is a constant of a wire, it is the resistance of a standard size wire of a material with 1 unit length and 1 unit cross sectional area, so 1m long with a cross sectional area of 1m.


I intend to investigate the constancy of resistivity; if resistivity is indeed a constant then it will be the same at every point along a chosen wire.


The standard deviation of the results can be calculated by using this equation,
By using the standard deviation of the results I predict that at every point tested along the chosen wire the resistivity will be within 2 standard deviations of the mean resistivity of the wire.


Approximately 97% of a sample will lie within 2 standard deviations of the mean, and approximately 99% of a sample lie within 3 standard deviations of the mean. Providing that all the results lie within 2 standard deviations of the mean, then it can be taken that resistivity is a constant.


Experimental Variables
The variables identified in this experiment are as follows,
Voltage
Current
Resistance
Length
Cross Sectional Area
Temperature
Throughout the experiment it will be assumed that there are no other variables present, and that temperature is constant at room temperature.


All the variables will be kept as close to constant as possible, barring length which will be the experimental variable and thus will be altered.


Experimental Procedure Theory
In order to calculate resistivity the voltage and current will be taken at a designated point along a wire, these values will then be used to calculate resistance using Ohms Law as shown,
This can then be combined with,
This will produce the equation,
The values used will be recorded and then this process repeated at a decided interval along a 1m length of wire.


Prior to this, the wire to use and the interval regularity must first be deiced, and safety precautions taken into account.


Preliminary Experiment
An array of wires were made available that varied in type and SWG (Standard Wire Gauge) values.


Copper SWG 30
Constanton SWG 28
SWG 30
Nickel Chrome SWG 18
SWG 22
SWG 30
In order to determine the appropriate wire for experimentation, a preliminary experiment was performed testing each wires resistive properties at 50cm length and 100cm length while using a power supply stated as being 6V with a supplied current of 1A.


The results were as follows,
Material
SWG
Length (cm)
PD (V)
Curent (A)
Resistance (?)
Copper
30
50
0.30
1.08
0.28
100
0.35
1.09
0.32
Constanton
28
50
2.55
1.06
2.40
100
4.55
0.93
4.89
30
50
3.30
1.03
3.20
100
5.10
0.83
6.10
Nickel Chrome
18
50
0.62
1.09
0.57
100
1.16
1.09
1.06
28
50
1.80
1.07
1.68
100
3.00
1.04
2.88
30
50
5.43
0.72
7.54
100
5.95
0.41
14.51
Theses results were then used to determine which wire would be most appropriate for use in the experiment.


Preliminary Conclusion
It was decided from the data supplied by the preliminary experiment that the most suitable wire for use in this investigation would be Constanton with an SWG value of 30.


At 50cm the resistance was calculated as being 3.20? and at 100cm the resistance was calculated as being 6.10? this demonstrates an almost 100% increase in resistance over the length of 100cm, thus showing that this wire has a wide range of resistive properties.


Although other wires tested exhibit similar properties, the only other wire to exhibit such a high increase across 100cm is Nickel Chrome (SWG 30) but this wire was ruled out due to the very high values of resistance which would require power supplies of much higher rating in order to collect accurate readings. The voltage required through a wire of that resistance would cause the wire to heat up rapidly presenting a possible safety issue.


The experimental wire is therefore Constanton (SWG 30).


Preliminary Diagram
A
V
Power Supply
100cm of Constant 30
Power Supply 6V, 1A
Banana Lead
Ammeter
Voltmeter
Experimental Procedure
100cm of Constanton will be connected in series with an ammeter and a 6V (1A) power supply. In addition a voltmeter will be attached in parallel. One terminal of the supply will be attached to one end and remain there, the other terminal will move up the wire at 10cm intervals until a full 100cm of wire has been tested.


A voltage and current reading will be taken at each interval and used to calculate the resistance using the equations stated in Experimental Procedure Theory on page 1.
The process shall then be repeated a further 2 times, thus collecting three values of voltage, current and resistance at each length. Upon completion of the full 3 tests, an average for each length will be taken, and then the resistivity calculated at each point.


Experimental Safety
During the experiment precautions were taken to ensure that the temperature of the wire remained safe, although it is to be expected that it will get hot and so a safe distance was maintained from the wire where possible when a current was flowing through it.


Only a 6V, 1A supply was used, in order to reduce chances of serious electrocution through either experimental or electrical fault.


All equipment was marked as being safe for use in an educational establishment by the presence of a valid PAT test certificate.


Apparatus Used
The apparatus used during the main experiment is the same as that used in the preliminary experiment and can be found in the diagram at the top of this page.


The preliminary experiment tested multiple wire types, the main experiment only tested Constanton SWG30.


Results
The table below contains the raw data collected during the main experiment on Constanton SWG30.


Run 1
Length (cm)
P.D. (V)
Current (A)
10
2.29
1.87
20
4.05
1.86
30
5.41
1.58
40
5.72
1.26
50
5.78
1.02
60
5.80
0.84
70
5.82
0.73
80
5.82
0.64
90
5.84
0.56
100
5.84
0.53
Run 2
Length (cm)
P.D. (V)
Current (A)
10
2.01
1.82
20
4.22
1.90
30
5.33
1.59
40
5.70
1.27
50
5.76
1.02
60
5.78
0.85
70
5.80
0.73
80
5.80
0.50
90
5.82
0.57
100
5.82
0.52
Run 3
Length (cm)
P.D. (V)
Current (A)
10
2.08
1.79
20
4.72
1.81
30
5.38
1.40
40
5.70
1.21
50
5.76
0.99
60
5.79
0.69
70
5.80
0.72
80
5.81
0.63
90
5.82
0.55
100
5.82
0.52
Analysis
From the raw data, a mean value for P.D and current was calculated for each length, and used to calculate a value for resistance at that point using Ohms Law,
This produced the following data,
Length (cm)
P.D (V)
Current (A)
Resistance (?)
10
2.13
1.83
1.16
20
4.33
1.85
2.34
30
5.37
1.52
3.54
40
5.70
1.26
4.58
50
5.77
1.01
5.71
60
5.79
0.79
7.36
70
5.82
0.73
8.02
80
5.81
0.59
9.97
90
5.82
0.56
10.41
100
5.83
0.52
11.41
The above data tells us a variety of basic things, we now know that as the length increases the current decreases, and the voltage and resistance increase.


This is shown in the graph above, where it can be clearly seen that the as the length of the wire increases, the voltage will sharply increase until it levels out at 40cm. The current however, decreases at an almost uniform rate as the length increases. In the graph above it can be seen that the resistance increases steadily as the length increases.
From the raw data collected the resistivity can be calculated by taking calculating resistivity at each point along the wire and then taking an average value for the wire as a whole.


First however the cross sectional area of the wire must be calculated, using the standard wire gauge values (SWG) of diameter, it can be calculated using pi that the cross sectional area is 0.08mm and so is 0.0008m.


This value for cross sectional area can now be used in the equation to work out resistivity,
This formula was used to calculate the resistivity at each point along the wire,
Length (cm)
Resistance (?)
Resistivity ( )
10
1.16
20
2.34
30
3.54
40
4.58
50
5.71
60
7.36
70
8.02
80
9.97
90
10.41
100
11.14
In order to prove or disprove the hypothesis the standard deviation must be calculated using the equation,
We know that,
Also,
And,
Using the equation, we can determine that standard deviation is equal too,
It is known that about 97% of the data lies within 2 standard deviations of the mean, therefore, providing all the data lies within 2 standard deviations of the mean then it can be considered to be a constant.


The highest value calculated was 9.97×10-4 and the minimum was 8.91×10-4
The mean equals,
The upper limit equals,
The lower limit equals,
This shows that,
Maximum < Upper Limit
Minimum > Higher Limit
Therefore, the data all lies within 2 standard deviations of the mean and allowing for variances in the experimental conditions, resistivity can be considered a constant value for Constanton and shall be taken as being 9.28×10-6?/m
Sources of Random and Methodological Error
Identified sources include a natural variance in the power supply, which due to conditions out the control of the experiment cause a fluctuation in the power supply and consequently it isnt a constant 230V supply, causing the outputted voltage of 6V to vary, along with the outputted current. Also another possible source of random error is natural impurities in the wire which could cause the flow of current to be affected thus effecting the collected data.


The most major source of error will be down to human error, such as inaccurate measurement of current and voltage during the experimental process, and inaccurate measurement of lengths, and placing of connections to the experimental wire from the power supply.


The use of standard deviation allows for these errors by giving a tolerance level of 2 standard deviations from the mean of the data collected.