This practical report outlines procedures for measuring the power of convex lenses and improvements to standard methods.

I have been given a box of 54 lenses with different powers, marked with different colours. My aim was to verify these given values, to find an improvement of my measurements and then to use this method to find the power of the lenses with unknown powers in the box and assemble them to the colour groups. The power has been calculated from measured values of u (distance from object to lens) and v (distance from focussed image to lens) by using the lens equation.

## TABLE OF CONTENT

1. Introduction and Theory

2. Aim

3. Experimental Procedure

3.1. First Experiment

3.2. Second Experiment

4. Results

4.1 Calculations

4.2 First Experiment

4.3 Second Experiment

4.4 Unknown Lenses

5. Conclusion

5.1 Definitions related to accuracy

5.2 Typical sources of imprecision and bias

5.3 The focal range

5.4 Final results

References and Picture Credits

Appendix

1. First experiment

2. Second experiment

## 1. Introduction and Theory

In this Quality of Measurement coursework the power of a set of convex lenses has been calculated from measured values of *u* (distance from object to lens) and *v* (distance from focussed image to lens) by using the lens equation.

The lens equation is rather simple: the incoming curvature plus the curvature added by the lens equals the curvature coming from the lens. Curvature in dioptres is calculated by taking the inverse of the distance from the source. The distance object-lens *u* must be negative, the distance image-lens *v * negative^{1}. Therefore:

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Figure 1: Diagram summing up the lens equation

Hence the power in dioptres can be calculated by taking the inverse of the focal length in metres.

## 2. Aim

I have been given a box of 54 lenses with different powers, marked with different colours. An overview is given in the table below:

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My aim was to verify these given values, to find an improvement of my measurements and then to use this method to find the power of the lenses with unknown powers in the box and assemble them to the colour groups.

## 3. Experimental Procedure

In all my experiments, I have worked with a fixed value of u and adjusted the position of the screen to give a focussed image to then measured the value of v. The value of u always had to be bigger than the focal length, otherwise it would give an imaginary picture (if *u* smaller than focal length) or focus at infinity (if *u* equals focal length).

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Figure 2: Set-up with lamp (filament bulb), image (an L shape on Figure 3: Lens at ruler, a black line in the middle of the lens cardboard), optical board (with millimetre ruler) and lens. holder made the reading easier and more accurate.

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Figure 4: The image has been projected on a white wooden Figure 5: Screen at ruler, the reading has been taken from the screen. left edge of the vertical board. (Reading: 21.6 cm).

For the object, I used a cardboard with an incised L with rough translucent paper glued behind it which made it easier to determine the focal point. An optical board (a wood rod with a 1.5 m ruler attached to it) has been used to measure the distances. Its scale is to the nearest millimeter, it has therefore an uncertainty of ± 0.05 cm.

For the precision of the results this is not the main source of uncertainty but the focal range. When trying to find the focal point, it is sometimes not possible to determine one single point where the image is focussed but a range of values within which it can be said that the image is focussed. To reduce this source of uncertainty in the second experiment was my aim after having discovered it in the first experiment. The uncertainty produced by the limited resolution of the ruler can be neglected for now as it is very small and a ruler with higher precision was not available. It will be discussed again in the conclusion.

### 3.1. First Experiment

At first, I measured one single value for each lens in the box and then calculated its focal length and its power. Then, I decided to take three more measurements for each lens so that I would have four values which I could then average and hence improve the reliability of my results. Moreover, I found the minimum and maximum measured value for each lens and hence calculated the minimum and maximum focal length and power for use in error bars on my diagrams.

### 3.2. Second Experiment

During my first Experiment, I noticed that if the value of *v* is large, there is a range of distances where the image could be said to be focussed. Instead of taking four rather random values within this range, I have decided to exactly determine this range with different values of u and observe wether there is a change in the uncertainty. If my assumption was true, an exacter value of the focal length and power could be obtained (see section 5.3).

Additionally, I measured the image height each time to be able to calculate the magnification. For this, I used a vernier calliper. It measures to the next hundredth of a centimeter so it has an uncertainty of ± 0.005 cm. By multiplying the hence calculated value of the magnification by the given value of u, I could determine an expected value of v.

## 4. Results

### 4.1 Calculations

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From the value of the focal length, I could calculate values for the Power, P in dioptres.

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Additionally, I have measured the object and image height to obtain a value for the magnification and hence to be able to calculate an expected value for *v*.

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To find averages and measures of spread, I have used the following statistical methods:

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I will show my calculation for the first experiments with the help of the example B1.

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### 4.2 First Experiment

The diagrams show the measured value of *v* and the value of P with respect to the lens number. The error bars show the PMAX-PMEAN on the positive end and PMEAN-PMIN on the negative end on the power diagrams and vMAX-vMEAN on the positive end and vMEAN-vMIN on the negative end on the diagrams showing the values of v.

For all colours, the y-axis has a range of 1D, apart from the black lenses because of the high fluctuation of values. The diagram for the values of v has a range of 30 cm on the y-axis.

The data tables and an explanation to them can be found in the Appendix.

1. Black Lenses

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Diagram 1a: The y-axis has a range of 5D. On this diagram there clearly are some outliers, lens 3 and 5 that should be in the white group.

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Diagram 2a: The y-axis has a range of 80cm. Again, the outliers can be identified quickly.

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Diagram 1b: The outliers B3 and B5 are now removed and the y-axis

now has a range of 3D. There is still much fluctuation. Lens B4, 7 and 14 are very much below the specified power in the box but cannot be assembled to other groups.

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Diagram 2b: The y-axis has now a range of 30 cm.

2. White Lenses

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Diagram 3: The y-axis has a range of 1D, therefore the error bars appear much larger than on the diagram for the black lenses. Here there are no outliers, lens W10 however has a slightly higher power than the others.

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Diagram 4.

3. Red Lenses

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Diagram 5: Lens R8 has a power slightly below the average but as the error bars are quite large, it may well be that this due to uncertainty.

Diagram 6.

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4. Green Lenses

Diagram 7: Only three green lenses were in stock.

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Diagram 8.

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5. Blue Lenses

Diagram 9: Again, only three lenses were available.

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Diagram 10.

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6. Orange Lenses

Diagram 11: The power given on the list is 20D, so these results are far off. This case will be discussed in section 5.2.

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Diagram 12.

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**[...]**

^{1} Dobson, Grace, Lovett: *Physics*, Collins Advanced Science, 1997