V = 5.004
Total resistance calculated from meter readings, from Ohms law =
= 50040Ω
Because there are no parallel branches for the current to flow in this section of the circuit we are considering, the current will always be the same as you give the resistor to start with plus the 40Ω from the ammeter. As shown in the above equation you are multiplying by the current and then diving by that same current to give the answer, and thus this cancels down to give the resistance.
Other Values obtained due to calculations:
As you can see, this circuit set-up is the better set up for higher resistances. You can see this more clearly if you look in the percentage change column, for 50Ω, the total resistance calculated by the meter readings is 80% off the original resistance value. But as the resistances get higher for example 40000Ω, the total resistance calculated by the meter readings is only 0.1% off the original resistance value.
Overall Observation from Theoretical Modelling
From my findings above I know that in Circuit Two, the resistance value will always be 40Ω off the resistance value, which is labelled on the resistor in the first place. I plotted a graph (on the next page) to see when the values of the theoretical resistances for both circuits would cross, because then I can find out for what range of resistances to use the first circuit and for what range to use the second circuit. The two lines on the graph meet at approximately 1435Ω, so I investigated further by conducting more calculations for circuit One, to find out at which value it would be the same as Circuit Two (40Ω off labelled resistance value):
As you can see from the calculated data, the first circuit has a 40Ω difference from the inputted value at around 1434Ω – 1435Ω. I can derive from this, that in theory at least, it would be better to use the first circuit for smaller resistances up to about 1435Ω and after this, Circuit Two would be more accurate. Then percentage change for both circuits would be the same at this point, but at values after this the percentage change for resistances in Circuit One, would increase, while for Circuit Two the percentage change would decrease.
Apparatus:
I will need the following for my practical experiment:
- A Voltmeter to the specification mentioned on the first page on my planning:
Maximum Pd 5V, Maximum Current 100μA
Until the actual experiment commences, I will not know for sure how accurate this device is but I will ensure that I pick the most accurate equipment that I have at my disposal. The same goes for the ammeter. Upon Investigating in various textbooks, I have found that digital voltmeters and ammeters are more accurate, so if possible I will endeavour to obtain one for use in my actual experiment.
- An Ammeter to the specification mentioned on the first page on my planning:
Maximum Current 2mA, Resistance 40Ω
- A Power Pack, on which I can vary the Voltage (and therefore current)
- A Set of labelled Resistors - quite a large range, from about 50Ω to 50000Ω. I decided not to use a rheostat because of its severe limitation – not being able to accurately read the resistance to start with. The resistors also need to be of a low tolerance level (how accurate the labels on the resistors are).
- Several Wires, all of the same material and of negligible resistance. I have decided to use Copper wires as they have a low resistance and resistivity and because they are widely available. I have heard of wires have almost no resistance but they have to be used at temperatures approaching absolute zero, because of this fact and the fact that there are very expensive, I have chosen copper because it suits my needs and its negligible resistance shouldn’t affect my readings.
- A holding station for the resistors – This will have a resistance but I consider it to be negligible so it shouldn’t be taken into account in my calculations.
Method
I plan to set Circuit One out in the way that is shown on the diagram above for a set resistance. I will then turn up the voltage (and therefore current) to an amount at which I can read from the two meters and then take note of these values I will then calculate the resistance from these two values using ohms law.
After this I will change the circuit, so that it becomes Circuit Two and then find the resistance of this circuit (using the above described method). This should hopefully give an insight into which of the above circuits is the most accurate.
Fair Test
By not moving the components and wires I will endeavour to keep the resistance constant, by not restricting the flow of current (and thus altering the resistance).
I will use the same resistor for each circuit because I don’t know how accurate the labels on the resistors are, and if it is changed the resistance of the other resistor could be different from the original.
While reading the voltmeter and ammeter (if they are analogue, not digital), I will stay directly above the meter so as to reduce parallax error, and hopefully this will enable my results to be more accurate.
If the voltmeter and ammeter are not zero, I will take this into account by subtracting this value off the appropriate value at the end. This is known as a zero error.
I will try to keep the wire of the resistors as straight as possible so to improve the flow of electrons and hopefully keep the experiment fair.
Another issue is heat, as the Power Pack (or Lab Pack) is left on, the temperature of the wires, and the resistors increases, and so, the longer I leave the power pack on for the more inaccurate my results get. This is because of Ohms law. Ohms law states “the voltage flowing through a metal is directly proportional to the current as long as the temperature remains constant”. This statement means that if the voltage were to double the current would also double. But if the temperature increases, the resistance of a metal increases, because the atoms vibrate more which makes it more difficult for the free electrons responsible for the current to flow. So by limiting the amount of time the Power pack is left on for I will try to keep the resistance constant.
I will keep all the equipment the same throughout – the ammeter, the voltmeter, the wires and the power supply. This is because the resistances of these control variables could vary from one to the other, so in keeping them the same I will be adding to the accuracy of my results.
Safety
I need to ensure that my experiment is conducted in the safest way possible, if not it could endanger lives, as well as damage equipment. My main concern is the Power Pack overheating or passing too large a current. To stop overheating I will limit the amount of time that the power pack is on for, as well as being safe it also doesn’t not increase the resistance if the temperature remains constant (see Fair Test, above).
The maximum Pd that the voltmeter can handle is 5V so I will keep the maximum emf at this value. This also ensures that the current in the wires is never enough to heat up the wires very significantly. The Power Pack has a cut-off switch if the current is too large, which is an added safety feature.
The normal rules of the lab will also be enforced such as not bringing any fluids (such as water) near electrical equipment as this could endanger lives.
Actual Method Used
When conducting my actual experiment to observe if my theoretical findings were actually correct, I noted the readings for Circuit One. Then I switched it off and moved the wire from the voltmeter into the ammeter to transform it into Circuit Two. I then switched the power pack off, and back on again, and repeated the same again for the same resistor, just to check that I hadn’t made an error in my initial observations. From these two results for each circuit I calculated an average resistance.
I used an analogue Voltmeter and instead of an ammeter I used a galvanometer (which is basically the same thing), because digital ones were not available. They were to the same specification mentioned in my plan:
Voltmeter - Maximum Pd 5V, Maximum Current 100μA
Ammeter - Maximum Current 2mA, Resistance 40Ω
I kept all the other basic equipment and the wires the same as mentioned in my plan.
To make my experiment fair:
- I didn’t move the components if I didn’t need to.
- I used the same resistor for each circuit.
- I reduced parallax error at all time by staying directly above the meters when taking readings.
- I kept the resistors as straight as possible to improve electron flow.
- I also kept the power pack on for the minimal amount of time so as not to heat the resistors and therefore alter the resistance.
- I made sure that the meters were at zero and if not I took this into account.
Results
The first column of this table shows the resistance that was labelled on the actual resistor. The second and third columns are the actual readings that I noted from the Voltmeter and ammeter. I have then calculated the resistance from these two readings in the fourth column.
First Set of Readings for Circuit One:
The first column of this table shows the resistance that was labelled on the actual resistor. The second and third columns are the actual readings that I noted from the Voltmeter and ammeter. I have then calculated the resistance from these two readings in the fourth column.
Second Set of Readings for Circuit One:
The first column of this table shows the resistance that was labelled on the actual resistor. The second and third columns are the actual resistance values that were taken in the two attempts I did. The fourth column is the average of these two readings.
Average Resistance for Circuit One:
The first column of this table shows the resistance that was labelled on the actual resistor. The second and third columns are the actual readings that I noted from the Voltmeter and ammeter. I have then calculated the resistance from these two readings in the fourth column.
First Set of Results for Circuit Two:
The first column of this table shows the resistance that was labelled on the actual resistor. The second and third columns are the actual readings that I noted from the Voltmeter and ammeter. I have then calculated the resistance from these two readings in the fourth column.
Second Set of Readings for Circuit Two:
The first column of this table shows the resistance that was labelled on the actual resistor. The second and third columns are the actual resistance values in the two experiments that I undertook. The fourth column shows an average of these two values.
Average Resistance for Circuit Two:
Instead of using percentage changes and resistance changes like I did in my original plan, I decided to ratio the two average resistances together. This is because I did not know if the labelled resistances were accurate. The tolerance of the resistors that were used was +-5%. This means that the actual resistance of the resistor can actually be as much as 5% above or below the labelled resistance, and for high resistors this is quite a high margin of error. For example:
For 33000Ω
Plus 5% = 34650
Minus 5% = 31350
This above example is 1650Ω off the labelled resistance value, this is quite high and, in retrospect, these tolerance levels are too high to be used in when accurate readings are needed.
Conclusions
As my results have show my initial calculations into which circuit was the better one for calculating resistances is true. Circuit One (the ammeter connected outside the voltmeter) is better for low resistances, up to a point, which is around 1435Ω as shown in my initial calculations. After, this circuit two becomes more accurate, and Circuit One’s accuracy decreases very rapidly as the resistances increase.
I repeated the experiment twice because I believed that this was necessary to increase the exactness of my results and this could also rule out some human error. The galvanometer and the voltmeter were in some cases hard to read and I was sometimes uncertain which value the meters were pointing at.
Taking note of the hand drawn graph before this page you can see that it is a graph of the resistance obtained from Circuit One against the Resistance obtained from Circuit Two. The graph is only plotted up to a labelled resistance of 2200Ω because that is all the scale would allow. I feel if I were to repeat this experiment I would take a wider range of results so I could plot them accurately on a graph. As you can see the graph has an almost perfect positive correlation. Except for one anomalous result. To find out how strong the correlation was I decided to calculate this using “the product moment correlation coefficient, r”. r can take values between +1 and –1 (+1 being a perfect positive correlation, -1 being a perfect negative correlation). This is calculated from the following formula:
or
These values were calculated from the averages table of resistances for both the circuits, and are for only the points drawn on the graph.
Σx2 = 18616773
Σy2 = 20746414
Σx = 13559
Σy = 14520
Σxy = 19641773
Therefore:
Sxy = 7336980.5
Sxx = 7126367.938
Syy = 7569514
r = + 0.9989626386
The result for r is astonishingly close to +1 considering the accuracy of the readings I took. This value of r shows that my results have an almost perfect positive correlation. I have no doubt that if the one anomalous result wasn’t there that the value of the product moment correlation coefficient would be even closer to +1.
Although at the time of drawing the afore mentioned graph, I did not realise that there could be another trend in the results such as a curve towards the end. I had heard of log graphs before and I knew they were to do with scaling axis, so I received further tuition from my Physics Teacher on how to draw these graphs.
I plotted the same graph before (with Resistance’s for Circuit One against Resistances for Circuit Two) but on a log scale and a trend became clear, the results actually curve on the graph, which is not apparent on the graph that I drew before. This shows to me that the ratio of resistances changes as the resistances become larger i.e. the change becomes bigger.
I have also drawn another graph of the ration of Circuit two to Circuit one against the log of Circuit two. I did this because I know that Circuit Two should always be about 40Ω off, so I consider this to be an accurate standard to plot against. The graph shows up quite a few anomalous results that the other graphs don’t, such as the last labelled resistance of 47000Ω. This point on the graph is well off the predicted trend. This could have arisen due to heating or to the build quality of the resistor itself. The straightened set of results at the bottom of the graph actually shows where both resistances are roughly the same accuracy, i.e. the intercept point as shown in my planning.
My results actually agree with my prediction, Circuit One becomes less accurate, the higher the resistances are. Circuit two however is less accurate for lower resistances because the 40Ω resistance of the ammeter has to be taken into account because it is connected inside the voltmeter.
There is one anomalous result, which can be clearly seen on the hand drawn graph, and this is for 1435Ω, which is where, in my prediction I proposed the intercept would be. I believe this result came about because I had to set up the circuit in a different way, because the resistors came in set value and not in the values that I wanted to use. I knew from my studies that the relationship connected resistors in series was:
Rs = R1 + R2 + … Rn
This means that I could connect resistors in a line to obtain the required resistance. What I didn’t foresee at the time was that the holding station of the resistors could have a big enough resistance to affect my calculations and because I used about 4 holding stations this was amplified. This is what I believe led to the anomaly on my graph, also because there were more wires in the circuit, this could have led to increased resistance also. I also remember of the tolerances on one of the resistors that I used was silver this means that it is -+10% which also added to the inaccuracy of my results.
In my averages table there is also three, what I consider to be anomalous results. These could have arisen due to the heating of the resistor due to the current flowing through it or because of the tolerance level of the resistor. This could also have happened because of the build quality of the resistor itself, or maybe the resistor metal was impure and therefore had increased resistance. These results are shown by the Circuit One \ Circuit Two ratios – because they do not follow the ‘normal’ trend.
I think the biggest factor in influencing my results was the accuracy of the voltmeter and the galvanometer. The galvanometer especially because it was measure in milliamps and this made a bigger difference on my results.
E.g.
For 10000Ω =
5V / 0.0005A = 10000Ω
If the current was only slightly smaller, the readings would changes quite a lot:
5V / 0.0004A = 12500Ω
This is quite a change of resistance for a 0.1milliamp difference.
Why Circuit One isn’t good for High Resistances
Circuit One isn’t good for high resistances due to the position of the ammeter. The voltmeter is only registering the potential difference across the resistor, but the ammeter is measuring the current through the resistor plus that draw by the voltmeter. If the resistance of the resistor is high say 50000Ω this means that the voltmeter is taking more current than if the resistor was say 20000Ω. It is the fact that the ammeter is measuring all the current instead of that just for the resistor, that makes this circuit inaccurate for high resistances.
Why Circuit Two isn’t good for Low Resistances
Circuit Two isn’t good for low resistances also because of the position of the ammeter but this time it is because the voltmeter is measuring the potential difference across both components. This means that the resistance of the ammeter is being taken into account when the two meters show their results. The error in the calculation could be reduced if the ammeter had a lower resistance.
To Improve the Circuits
I think that after my working with this practical I reflect that the best circuit would be circuit two, but a certain amount of changes would have to be made to make it more reliable and accurate. Firstly, the resistance of the ammeter would have to be significantly reduced so as not to affect the final calculations, and secondly the voltmeter should have an almost infinite resistance so as not to take up any current and therefore alter the readings.
Evaluation
After conducting my experiment and analysing the results I feel that there is a great deal of improvement. My results did fit my prediction to a certain extent, but there were a few anomalous results, which would need to be re-checked if I were to conduct my experiment again. If I had known at the time of my experiment that anomalous results had occurred I would have gone back to check again, but unfortunately I could not repeat this due to time constraints.
Many factors could improve the overall accuracy of the investigation if I were to repeat this experiment I fell that I could improve on this accuracy, now that I know the good and the bad points of the investigation.
I feel that more results should have been taken by me to prevent the occurrence of anomalous results, even though I know that not all of them can be helped, I believe that some of them could be eradicated by putting a little more care and attention into taking readings and interpreting the scales of the meters. Repeating the experiment more time say at least 5 times would be much better to take the mean resistance for and lead to more precise results, and these results would be better to base conclusions on.
I also deduce that if I had taken a wider range of resistor values at regular intervals, I could have obtained an overall grasp of the results and this would have made interpreting them much easier. This would also give me more points to pot on graphs and therefore obtain a more accurate view of the experiment as a whole.
The accuracy for the resistors inconvenienced me throughout the experiment, the tolerance levels for the resistors was generally around 5%, and I think this is quite a big margin of error. It inconvenienced me because I did not have an accurate set of resistor values to do comparisons with. But basically that is what the whole experiment was about – how to calculate resistance. I realise now that the labelled resistances were just a guide to give an estimate of the value of the resistor.
The wires and holding stations of the resistors had, a resistance and I think this could have affected my results overall, especially when I connected resistors in series. If I were to repeat the experiment again I would try to get wires of a lower resistances, and resistors of the values that I need.
The Percentage error was different for each individual reading, below is a table of average percentage error for each labelled resistance.
As you can see some of the above percentage errors are colossal, and so this could have had a profound implication on my results. If I were to repeat this experiment again I would set the voltage and the current to their highest possible values so as to reduce error. I tried to do this in my experiment as can be seen in the above table but, sometimes it was just below, so I could read it more accurately.
The thing that bothered me most during the experiment was the accuracy of the two meters. They were on the whole very inaccurate. Parallax error also could have influenced my results and if this happened the resulting resistance could have been a long way off its actual value.
I think that digital meters would be have been a far better alternative to analogue, because they have a higher resistances, and human error could have been drastically reduced because the readout is digital. Parallax error would have been totally eradicated, and the reading would have been accurate to more decimal places.
Internal resistance of the components in question could have influenced my results, but I don’t think this happened because overall the readings were generally good.
I think overall the accuracy of my investigation was good, but there are numerous small factors that could have influenced a few of the anomalous results.
If I were to repeat the experiment again, I would try to find resistors that had a greater tolerance accuracy e.g. tolerances of 1% to get better resistor values. I would leave the power supply on for only a limited amount of time to prevent heating, which could change the resistance. I would like to use digital meters to record the current and voltage with. This would ensure that the results would be more accurate.
Overall though, I think my investigation did prove what I was trying to find and I could make conclusions that matched up with my evidence and my predictions. It must have meant that my results were fairly accurate and my investigation proved what I was out to find - You should connect the voltmeter without the ammeter in for low resistances up to 1435Ω for this particular circuit, but after this circuit two would give more accurate results.
