∴ Mean focal length f of the convex lens = (9.878 ± 0.122) cm
Method (b): Lens displacement method (for a convex lens)
Mean focal length f of the convex lens =
= 9.761 cm
Uncertainty
= −−−−−
= 0.154 cm
∴ Mean focal length f of the convex lens = (9.761 ± 0.154) cm
Method (c): Lenses-mirror method (for a concave lens)
Mean focal length f of the concave lens =
= 15 cm
Uncertainty = −−−−−
= 1.2 cm
∴ Mean focal length f of the concave lens = (15 ± 1.2) cm
Method (d): Lenses combination method (for a concave lens)
Focal length fu of the combined lens calculated using (1/u)-intercept =
= 17.391 cm
Focal length fv of the combined lens calculated using (1/v)-intercept =
= 14.815 cm
Mean focal length f of the combined lens =
= 16.103 cm
Uncertainty of the mean focal length f of the combined lens = −
= 1.288 cm
∴ Mean focal length f of the combined lens = (16.103 ± 1.288) cm
Focal length f1 of the convex lens = (9 ± 1) cm (calculated from method (a))
Focal length f2 of the concave lens = −
= ×−
= −20.404 cm
% error in f2 = % error in f1 + % error in f
= [( ) + ( )] × 100%
= 19.110%
Uncertainty of f2 = (−20.404) × 19.110%
= −3.899 cm
∴ Mean focal length f2 of the concave lens = (20.404 ± 3.899) cm
Percentage Errors
Method (a),
Mean focal length of the convex lens = (9.878 ± 0.122) cm
Percentage error = × 100% = 1.234%
Method (b),
Mean focal length of the convex lens = (9.761 ± 0.154) cm
Percentage error = × 100% = 1.578%.
Method (c),
Mean focal length of the concave lens = (15 ± 1.2) cm
Percentage error = × 100% = 8%
Method (d),
Mean focal length f2 of the concave lens = (20.404 ± 3.899) cm
Percentage error = × 100% = 19.109%
Error Analysis:
- Systematic error due to the use of metre rule to measure.
- Random error due to the parallax when reading the scale.
- More trials should be done to get more accurate measurement.
- Systematic error due to the transparency of the lenses used.
Discussion:
The percentage error of the measured value obtained by method (a) is 1.234%, and the one obtained by method (b) is 1.578%. When compare the percentage errors of the measured values of the focal length of the convex lens obtained by methods (a) and (b), the measured values obtained by method (a) is more accurate; its percentage error is much less than that of measured values obtained by method (b).
Method (a) is easy to set up, only a few things need to be adjusted when the experiment is proceeding. However, a lot of works need to be done, such as plotting graph, reading graph and a lot of calculation. This may make errors occurs more easily.
Although method (b) is more complicated than method (a), it is still worth to use. No graph and less calculation need to be plotted and done to find out the focal length, this will make fewer errors occur. However, more time is needed to be spent to set up and finish the experiment, as there are many apparatus used in this method. Moreover, it may be too difficult for students to adjust the apparatus to the right place and get the required value.
Therefore method (a): “Object & image distance method” is a fast accurate and easy method to measure the focal length of the convex lens.
The percentage error of the measured value obtained by method (c) is 8%, and the one obtained by method (d) is 19.109%. When compare the percentage errors, the percentage error of the measured value obtained by method (c) is more accurate than the other one.
Method (c) is quite hard to master. It is so hard to find where the concave lens should be put and also the plane mirror. After finish the first step of this method, a concave lens and plane mirror are to be placed in between the convex lens and the screen, but so much time needs to be spent to find the suitable positions for both apparatus. In this method, no graph and less calculation are needed to measure the mean focal length of the concave lens. Therefore, fewer errors occur and the measured value is more accurate.
The advantages and disadvantages of method (d) are similar to those of method (a). Method (d) only need to be adjusted a few things, it is easy to be done. However, there are much more works. In method (d), a graph is needed to be plotted, need to read from the graph to get some readings for further calculation, and then only the mean focal length of the combined lens is found. The focal length of the convex lens and that of the combined lens are used to find the focal length of the concave lens afterwards. After estimating the uncertainty, the mean focal length of the concave length is found finally. There are a lot of errors as there is so much calculation.
Therefore, method (c): “Lenses-mirror method” should be used as its percentage error is the least.
In daily life, the above methods can be used. For example, if people need to determine the focal length of the lens of a camera, they can use method (b): “Lens displacement method” to measure the focal length. Cameras are made used of two convex lenses to make the light rays converge, so that photographs can be taken. So, cameras have the apparatus needed for method (b). To carry out this experiment, the experimentalist has to hold his camera far away from the object and adjust the focus, once the image is sharp, measure the distance between the camera and the object. Then he needs to walk toward the object without adjusting the focus, once the image is sharp, measure the distance between the camera and the object again. Repeat the measurement to obtain more pairs of distances, and then just follow the procedure. The focal length of the lens of a camera then can be found by method (b): “Lens displacement method”.
Conclusion:
Method (a): “Object & image distance method” is suggested to measure the focal length of a spherical convex lens. Method (c): “Lenses-mirror method” is suggested to measure the focal length of a spherical concave lens.
