3. 1 Aims and methods
The aim of this investigation was to determine the power of a selection of lenses.
1.1 Method
I planned to do this by measuring the distance from lens to light source (u) and the distance from lens to
image (v). I would then use the lens makers formula to determine the focal length (f) of the lens, and then
find the power (P) from that value.
The lens makers formula is 1/f = 1/v + 1/u and the formula for power is P = 1/f .
I set up my apparatus as shown below.
For each lens I started with the middle of the bulb adjacent to the 100cm marking on the rule. I then
moved the screen back from the lens until I could see the filament of the bulb in focus. I marked this down
as the lower bound of v. After this, I moved the screen further away from the lens, until the image of the
filament was just out of focus. I then took this as the upper bound of v. When a measurement was done,
with an upper and lower bound recorded, I moved the bulb and holder forwards 5cm, and repeated. When
the value of v went over 100cm, I moved onto the next lens.
1.2 Alternative Methods
There are several other methods to measure the focal length of a lens, and thereby measure the power. One
method would have been ray tracing, which would have had much smaller error margins, though due to a
lack of equipment, it would not have been possible for me to use that method.
2 Properties of instruments
The only instrument that I used to take measurements that were used in the final calculation were a pair of
meter rules. For all intents and purposes, they were identical. I did also use my eyes to determine when the
image from the lens was in focus, though it would be impractical to correct any errors found in them, so I
tried to compensate for this lack by taking two measurements, the upper bound, where I thought that the
image was just past coming into focus, and the lower bound, where I thought that the image was just before
going out of focus.
2.1 Resolution
The smallest marked interval on the rules was 1 mm, and that was the precision I took measurements to. In
general, I quoted all numerical values to 1 decimal place.
2.2 Sensitivity
As a rule is a linear instrument, it has a sensitivity independent of its input, thus this was not something I
needed to worry about.
2.3 Calibration
It was not practical to calibrate the rules, as that would require equipment we did not have available. It
would also have been rather impractical.
2.4 Response time
Response time for a rule is instantaneous.
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4. 2.5 Stability
I secured the rules with blutack to try and increase stability. The rules were mounted across two tables,
which decreased the stability, as if one of the tables was moved, the rule would move, changing the angle
between the two rules from 180o , and thus changing the distance.
2.6 Zero Error
I checked for zero error on the rules by placing them with their zero ends together, and measuring the distance
between the 1cm markings on both of them. If it did not equal 2cm, then there was a zero error, and I would
test each ruler separately to determine the zero error on them. However, it did equal 2cm, and I could safely
say that there was no zero error.
3 Uncertainties
The percentage errors for the final calculation of P was generally from 3 to 6 percent. This was not too
bad, though I did find that some results had anomalous percentage errors, though having an actual reading
roughly in line with the rest of the results. I left these results out of the final calculations.
3.1 Measuring errors on v
As previously stated, I took two readings, an upper bound and a lower bound for when the image came in
and out of focus. I then used the mean of these two results as my actual result. Looking back, it might have
been a good idea to take a third reading, where I deemed the picture to be most in focus, and use this as my
actual result. However, this would have made calculating percentage errors much more complicated, as the
error would have had a different value for the positive and negative error.
3.2 Paralax Errors
There were several parallax errors I found, and I noted all of these down. I also found one systematic parallax
error, from the fact that I was taking the measurement of distance from the other side of the screen that
the image was projected on, meaning that the width of the screen was not being taken into account. This
meant that all of the results for v were 1mm larger than I had recorded. I corrected this error when I came
to process the data.
3.3 Miscellaneous errors
There were some errors I did not take into account, which I believe may account for some of the anomalous
results I encountered.
3.3.1 Lack of equipment
There was some specific equipment I had requested that I did not receive, which I believe resulted in some of
the errors. One item in particular that I did not have was a lens holder; I had to improvise with a triangular
piece of metal I was given, and some blutack. This meant that the lens became susceptible to unintended
changes in angle, which would have had serious effects on the reading. However, it was all but impossible
to measure the angle of the lens constantly, and I had to resort to just checking the alignment after every
reading, which was rather suboptimal.
3.3.2 Equipment instability
Both the rules were made of wood, which is expands proportionaly to humidity in the local atmosphere. I
did not account for changes in humidity in my results however, as I belive that the inacuracies gained via
the expansion of the rules would be neglible.
4 Risks
This was a relatively safe experiment, as there was not much that could go wrong. However, I identified the
following risks:
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5. Risk Probability (1..10) Severity (1..10)
Splinter from rules could be come embedded in finger 3 1.5
Lens breaks and shatters creating dangerous glass shards 2 5
Blu tack becomes tangled in hair causing irritation 1 9
Someone could poke themselves in the eye with the end of a meter rule 2 7
A paper cut could be gained from the paper used to take notes 3 5
The bulb of the lamp might break, creating dangerous fragments 1 6
I took appropriate measures to minimise these risks, i.e. Not dropping lenses and not putting blu tack in
hair.
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6. 5 Results
I will go through each set of results separately.
5.1 Lens labelled f = 25cm
This was the first test I did, and it went without incident. The percentage error on the values of p ranged
from 0.5% to 6%, which I thought was respectable.
As we can see in Fig 2, the value of f is definitely near 25cm, with only minor deviations from a mean
value of 25.1cm. Fig 3 is a plot of Power against U, and the same trend is evident; small deviations in P from
a mean value of 39.8 dioptre. The size of the error bars may be disconcerting, however the mean percentage
error is 4.2%. The box plot of the power shown in Fig 1 illustrates quite how tightly the values of P are
distributed.
I found 39.8 dioptre as the value of P for this lens, with a percentage error of 4.2%.
Figure 1: Boxplot of P for lens labelled f = 25cm
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7. Figure 2: Graph of F against U for lens labelled f = 25cm
Figure 3: Graph of P against U for lens labelled f = 25cm
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8. 5.2 Lens labelled f = 20cm
This experiment was fairly nondescript, with the exception of a single point. If we look at Fig 5, the points
have a relatively uniform error margins, except for the point u = 30cm, which has a percentage error of 34%.
The point itself deviates from the mean by just over 2.5%, yet due to the abnormally large error margin, I
discarded the result when calculating mean values.
I found 50.6 dioptre as the value of P for this lens, with a percentage error of 4.9%.
Figure 4: Boxplot of P for lens labelled f = 20cm
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9. Figure 5: Graph of F against U for lens labelled f = 20cm
Figure 6: Graph of P against U for lens labelled f = 20cm
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10. 5.3 Lens labelled f = 15cm
As you can see from Fig 7, this experiment produced a much looser distribution of results. Despite this,
there are some definite anomalous results, in particular, u = 40cm and u = 70cm. I left these two results
out, as I felt they deviated too far from the mean. I believe the reason that the result for u = 70cm was
anomalous was because the light source had been jogged, resulting in an incorrect u value, which was then
carried through the calculations.
I found 66.2 dioptre as the value of P for this lens, with a percentage error of 5.7%.
Figure 7: Boxplot of P for lens labelled f = 15cm
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11. Figure 8: Graph of F against U for lens labelled f = 15cm
Figure 9: Graph of P against U for lens labelled f = 15cm
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12. 5.4 Lens labelled f = 10cm
The results for this experiment were rather erratic. I found 5 results that I thought were anomalous. They
were: u = 15cm, u = 20cm, u = 25cm, u = 40cm, u = 100cm. I believe many of these are likely to be for the
same reason as for the uncertainties with the lens labelled f = 15cm; the u distance was measured incorrectly.
This would account for why all the results from 15 through to 25 are anomalous, as I made the mistake of
sitting down when taking these measurements.
I found 100.2 dioptre as the value of P for this lens, with a percentage error of 1.9%.
Figure 10: Boxplot of P for lens labelled f = 10cm
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13. Figure 11: Graph of F against U for lens labelled f = 10cm
Figure 12: Graph of P against U for lens labelled f = 10cm
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14. 5.5 Lens labelled f = 5cm
This experiment with quite a tight distribution of values for power. However, there were still two outliers,
both of which I consider to be anomalies. They were the values for u = 15cm and u = 20cm. I think the
reason that these were measured incorrectly (for that is what I believe caused the anomaly) was partly due
to the fact that I felt that larger values fitted the pattern I had seen with previous lenses, and I therefore
read the measurements as slightly bigger than they were in reality.
I found 203.1 dioptre as the value of P for this lens, with a percentage error of 3.0%.
Figure 13: Boxplot of P for lens labelled f = 5cm
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15. Figure 14: Graph of F against U for lens labelled f = 5cm
Figure 15: Graph of P against U for lens labelled f = 5cm
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