Determination of the Focal Length of a Convex Lens
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- Shaikh Ilyas
AIM: To determine the focal length of converging lens and it’s radius of curvature.
HYPOTHESIS: The relationship between u and v and the focal length f for a convex lens is given by . Where f is the focal length, u is the distance between the object and the lens v is the distance between the image and the lens. Real and Virtual Images: Lenses produce images by refraction that are said to be either real or virtual.
- Real images are created by the convergence of rays and can be projected onto a screen; real images form on the side of the lens that is opposite to the object and by convention have a positive image distance value;
- Virtual images are formed by the apparent extrapolation of diverging rays and cannot be formed on a screen, whereas virtual images form on the same side of the lens as the object and have a negative image distance value.[1]
BACKGROUND: For a thin double convex lens,refractionacts to focus all parallel rays to a point referred to as the principal focal point. The distance from the lens to that point is the principal focal length f of the lens. Below is the derivation of the lens formula
Following graphic illustrates a simple lens model:
where,
h= height of the object
h’= height of the object projected in an image
G and C = focal points
f= focal distance
u= Distance between the object and the focal point
O= Centre of the lens
v= Distance between the centre of the lens and image plane
Assumptions
- Lens is very thin
- Optical axis is perpendicular to image plane
Proving is true.
Proof
In ΔAHO,
In ΔEDO,
∴
----- (1)
In ΔBOC,
In ΔEDC,
∴
------ (2)
Equating equations (1) and (2),
Dividing both sides by v,
Hence the formula is proved.
VARIABLES:
Independent: Distance between the candle and the lens
Dependent: Distance (v) from the image to the lens
Control:
- This experiment was conducted in an almost dark room.
- Same sheet of paper used as the screen.
- A stable candle flame
- The time taken for a sharp and focused image to settle
- The size of the candle.
METHOD FOR CONTROLLING VARIABLES: Made sure that the room was sufficiently dark enough to carry out this experiment as smoothly as possible without any entrance of light from the outside. So I pulled down the blinds of the windows and also made sure that there was no draught present in the room that can make the candle flame unstable. Moreover, I waited for around 6-7 seconds for the image to be seen as sharp and focused. And throughout this experiment I used candles of the same make and size.
APPARATUS REQUIRED:
- 2 meter rules
- A white screen
- Candle
- Convex lens
PROCEDURE:
I divided this experiment in to 2 parts, A and B. In part A, I experimented using a single lens at a time, while in part B, I used 2 lens in contact at a time.
Part A:
- Firstly I set up the apparatus as shown in Figure 1 above by making the distances v and u the same. So the image observed on a plain white screen was focused and clear
- Recorded the value of the lengths u and v and thereby marking these original points using a chalk on the bench.
- Then I adjusted the length of u by moving it away from the lens by 5cm. Consequently, I adjusted the length of v until a sharp and focused image was seen.
- Recorded this distance of u and v
- Repeated step 3 - 4 for 7 different values of u by increasing the distance by 5 cm in each step. And recorded the values of u and v for every increment.
- Then I placed the candle and the screen back in their original marked positions.
- Finally, repeated the steps 1-8 by using different convex lenses A, B, C, D and E.
Figure 1: Setup of the apparatus for Part A
Part B:
- Firstly I set up the apparatus as shown in Figure 2 by making the distances v and u the same. So the image observed on a plain white screen was focused and clear
- Recorded the value of the lengths u and v and thereby marking these original points using a chalk on the bench.
- Then I adjusted the length of u by moving it away from the lens by 5cm. Consequently, I adjusted the length of v until a sharp and focused image was seen. Recorded this distance of u and v
- Repeated step 3 - 4 for 4 different values of u by increasing the distance by 5 cm in each step. And recorded the values of u and v for every increment.
- Repeated the above steps 1-5, thrice.
Figure 2: Setup of the apparatus for Part B
DATA COLLECTION AND PROCESSING:
- Part A:
Table 1: Data collected for convex lens A
u (distance between the lens and candle)+ 0.1cm |
v (distance between the lens and screen)+ 0.1cm |
15.0 |
25.1 |
20.0 |
21.5 |
25.0 |
17.0 |
30.0 |
14.7 |
35.0 |
14.2 |
40.0 |
13.6 |
45.0 |
13.0 |
Table 2: Data collected for convex lens B
u (distance between the lens and candle)+ 0.1cm |
v (distance between the lens and screen)+ 0.1cm |
15.0 |
28.9 |
20.0 |
24.2 |
25.0 |
19.2 |
30.0 |
15.8 |
35.0 |
13.9 |
40.0 |
13.2 |
45.0 |
12.7 |
Table 3: Data collected for convex lens C
u (distance between the lens and candle)+ 0.1cm |
v (distance between the lens and screen)+ 0.1cm |
15.0 |
24.6 |
20.0 |
21.1 |
25.0 |
16.5 |
30.0 |
14.3 |
35.0 |
13.9 |
40.0 |
13.4 |
45.0 |
12.9 |
Table 4: Data collected for convex lens D
u (distance between the lens and candle)+ 0.1cm |
v (distance between the lens and screen)+ 0.1cm |
15.0 |
28.7 |
20.0 |
23.6 |
25.0 |
17.4 |
30.0 |
14.9 |
35.0 |
14.0 |
40.0 |
13.4 |
45.0 |
13.0 |
Table 5: Data collected for convex lens E
u (distance between the lens and candle)+ 0.1cm |
v (distance between the lens and screen)+ 0.1cm |
15.0 |
25.8 |
20.0 |
20.1 |
25.0 |
15.4 |
30.0 |
14.3 |
35.0 |
13.9 |
40.0 |
13.1 |
45.0 |
12.5 |
- Part B:
Table 6: Data collected for Trial 1
u (distance between the lens and candle)+ 0.1cm |
v (distance between the lens and screen)+ 0.1cm |
30.0 |
60 |
40.0 |
38 |
50.0 |
33 |
60.0 |
30.1 |
Table 7: Data collected for Trial 2
u (distance between the lens and candle)+ 0.1cm |
v (distance between the lens and screen)+ 0.1cm |
30.0 |
58.7 |
40.0 |
37.8 |
50.0 |
32.6 |
60.0 |
30 |
Table 8: Data collected for Trial 3
u (distance between the lens and candle)+ 0.1cm |
v (distance between the lens and screen)+ 0.1cm |
30.0 |
61.5 |
40.0 |
38.7 |
50.0 |
33.2 |
60.0 |
29.6 |
Using the formula, R = 2f I can calculate the value for the radius of curvature. The value of f can be found using the equation.
- Part A:
Table 9:Data processing for convex lens A
u (distance between the lens and candle) + 0.1cm |
v (distance between the lens and screen) + 0.1cm |
Focal length (f) (cm) |
Radius of curvature (R) (cm) |
(f-x) |
(f-x)^{2} |
15 |
25.1 |
9.39 |
18.78 |
-0.62 |
0.38603 |
20 |
21.5 |
10.36 |
20.72 |
0.35 |
0.12328 |
25 |
17.0 |
10.12 |
20.24 |
0.11 |
0.01182 |
30 |
14.7 |
9.87 |
19.73 |
-0.14 |
0.02090 |
35 |
14.2 |
10.10 |
20.20 |
0.09 |
0.00833 |
40 |
13.6 |
10.15 |
20.30 |
0.14 |
0.01930 |
45 |
13.0 |
10.09 |
20.17 |
0.08 |
0.00576 |
Mean(f) = 10.01 |
Standard deviation: δm = = = 0.30967
Therefore, the focal length is 10.01+ 0.31 cm
The % error = = 3.1%
Table 10:Data processing for convex lens B
u (distance between the lens and candle) + 0.1cm |
v (distance between the lens and screen) + 0.1cm |
Focal length (f) (cm) |
Radius of curvature (R) (cm) |
(f-x) |
(f-x)^{2} |
15 |
28.9 |
9.87 |
19.75 |
-0.38 |
0.14761 |
20 |
24.2 |
10.95 |
21.90 |
0.69 |
0.47792 |
25 |
19.2 |
10.86 |
21.72 |
0.60 |
0.36098 |
30 |
15.8 |
10.35 |
20.70 |
0.09 |
0.00818 |
35 |
13.9 |
9.95 |
19.90 |
-0.31 |
0.09612 |
40 |
13.2 |
9.92 |
19.85 |
-0.33 |
0.11162 |
45 |
12.7 |
9.90 |
19.81 |
-0.35 |
0.12548 |
Mean(f) = 10.26 |
Standard deviation: δm = = = 0.47044
Therefore, the focal length is 10.26+ 0.47 cm
The % error = = 4.6%
Table 11:Data processing for convex lens C
u (distance between the lens and candle) + 0.1cm |
v (distance between the lens and screen) + 0.1cm |
Focal length (f) (cm) |
Radius of curvature (R) (cm) |
(f-x) |
(f-x)^{2} |
15 |
24.6 |
9.32 |
18.64 |
-0.57 |
0.32564 |
20 |
21.1 |
10.27 |
20.54 |
0.38 |
0.14350 |
25 |
16.5 |
9.94 |
19.88 |
0.05 |
0.00259 |
30 |
14.3 |
9.68 |
19.37 |
-0.20 |
0.04197 |
35 |
13.9 |
9.95 |
19.90 |
0.06 |
0.00361 |
40 |
13.4 |
10.04 |
20.07 |
0.15 |
0.02209 |
45 |
12.9 |
10.03 |
20.05 |
0.14 |
0.01879 |
Mean(f) = 9.89 |
Standard deviation: δm = = = 0.30500
Therefore, the focal length is 9.89+ 0.31 cm
The % error = = 3.1%
Table 12:Data processing for convex lens D
u (distance between the lens and candle) + 0.1cm |
v (distance between the lens and screen) + 0.1cm |
Focal length (f) (cm) |
Radius of curvature (R) (cm) |
(f-x) |
(f-x)^{2} |
15 |
28.7 |
9.85 |
19.70 |
-0.29 |
0.08633 |
20 |
23.6 |
10.83 |
21.65 |
0.68 |
0.46324 |
25 |
17.4 |
10.26 |
20.52 |
0.11 |
0.01308 |
30 |
14.9 |
9.96 |
19.91 |
-0.19 |
0.03595 |
35 |
14.0 |
10.00 |
20.00 |
-0.15 |
0.02105 |
40 |
13.4 |
10.04 |
20.07 |
-0.11 |
0.01158 |
45 |
13.0 |
10.09 |
20.17 |
-0.06 |
0.00346 |
Mean(f) = 10.15 |
Standard deviation: δm = = = 0.32524
Therefore, the focal length is 10.15+ 0.33 cm
The % error = = 3.2%
Table 13:Data processing for convex lens E
u (distance between the lens and candle) + 0.1cm |
v (distance between the lens and screen) + 0.1cm |
Focal length (f) (cm) |
Radius of curvature (R) (cm) |
(f-x) |
(f-x)^{2} |
15 |
25.8 |
9.49 |
18.97 |
-0.28 |
0.07574 |
20 |
20.1 |
10.02 |
20.05 |
0.26 |
0.06992 |
25 |
15.4 |
9.53 |
19.06 |
-0.23 |
0.05327 |
30 |
14.3 |
9.68 |
19.37 |
-0.08 |
0.00586 |
35 |
13.9 |
9.95 |
19.90 |
0.19 |
0.03548 |
40 |
13.1 |
9.87 |
19.74 |
0.11 |
0.01159 |
45 |
12.5 |
9.78 |
19.57 |
0.02 |
0.00049 |
Mean(f) = 9.76 |
Standard deviation: δm = = = 0.20508
Therefore, the focal length is 9.76 + 0.20508 cm
The % error = = 2.1%
- Part B:
Table 14: Data processing for Trial 1
u (distance between the lens and candle) + 0.1cm |
v (distance between the lens and screen) + 0.1cm |
Focal length (f) (cm) |
Radius of curvature (R) (cm) |
(f-x) |
(f-x)^{2} |
30 |
60.0 |
20.00 |
40.00 |
0.15 |
0.02168 |
40 |
38.0 |
19.49 |
38.97 |
-0.37 |
0.13366 |
50 |
33.0 |
19.88 |
39.76 |
0.03 |
0.00072 |
60 |
30.1 |
20.04 |
40.09 |
0.19 |
0.03672 |
Mean(f) = 19.85 |
Standard deviation: δm = = = 0.43905
Therefore, the focal length is 19.85 + 0.44cm
The % error = = 2.2%
Table 15: Data processing for Trial 2
u (distance between the lens and candle) + 0.1cm |
v (distance between the lens and screen) + 0.1cm |
Focal length (f) (cm) |
Radius of curvature (R) (cm) |
(f-x) |
(f-x)^{2} |
30 |
58.7 |
19.85 |
39.71 |
0.10 |
0.00961 |
40 |
37.8 |
19.43 |
38.87 |
-0.32 |
0.10300 |
50 |
32.6 |
19.73 |
39.47 |
-0.02 |
0.00047 |
60 |
30.0 |
20.00 |
40.00 |
0.24 |
0.05984 |
Mean(f) = 19.76 |
Standard deviation: δm = = = 0.16976
Therefore, the focal length is 19.76 + 0.17 cm
The % error = = 0.9%
Table 16: Data processing for Trial 3
u (distance between the lens and candle) + 0.1cm |
v (distance between the lens and screen) + 0.1cm |
Focal length (f) (cm) |
Radius of curvature (R) (cm) |
(f-x) |
(f-x)^{2} |
30 |
61.5 |
20.16 |
40.33 |
0.26 |
0.06875 |
40 |
38.7 |
19.67 |
39.34 |
-0.23 |
0.05387 |
50 |
33.2 |
19.95 |
39.90 |
0.05 |
0.00252 |
60 |
29.6 |
19.82 |
39.64 |
-0.08 |
0.00645 |
Mean(f) = 19.90 |
Standard deviation: δm = = = 0.14809
Therefore, the focal length is 19.90 + 0.15 cm
The % error = = 2.2%
CALCULATIONS AND DATA PRESENTATION:
Table 17: Data presentation for Convex lens A
(cm)^{-1} |
(cm)^{-1} |
(cm)^{-1} |
0.07 |
0.04 |
0.11 |
0.05 |
0.05 |
0.10 |
0.04 |
0.06 |
0.10 |
0.03 |
0.07 |
0.10 |
0.03 |
0.07 |
0.10 |
0.03 |
0.07 |
0.10 |
0.02 |
0.08 |
0.10 |
Figure 3: Graph of focal length for lens A
Table 18: Data presentation for Convex lens B
(cm)^{-1} |
(cm)^{-1} |
(cm)^{-1} |
0.07 |
0.03 |
0.10 |
0.05 |
0.04 |
0.09 |
0.04 |
0.05 |
0.09 |
0.03 |
0.06 |
0.10 |
0.03 |
0.07 |
0.10 |
0.03 |
0.08 |
0.10 |
0.02 |
0.08 |
0.10 |
Figure 4: Graph of focal length for lens B
Table 19: Data presentation for Convex lens C
(cm)^{-1} |
(cm)^{-1} |
(cm)^{-1} |
0.07 |
0.04 |
0.11 |
0.05 |
0.05 |
0.10 |
0.04 |
0.06 |
0.10 |
0.03 |
0.07 |
0.10 |
0.03 |
0.07 |
0.10 |
0.03 |
0.07 |
0.10 |
0.02 |
0.08 |
0.10 |
Figure 5: Graph of focal length for lens C
Table 20: Data presentation for Convex lens D
(cm)^{-1} |
(cm)^{-1} |
(cm)^{-1} |
0.07 |
0.03 |
0.10 |
0.05 |
0.04 |
0.09 |
0.04 |
0.06 |
0.10 |
0.03 |
0.07 |
0.10 |
0.03 |
0.07 |
0.10 |
0.03 |
0.07 |
0.10 |
0.02 |
0.08 |
0.10 |
Figure 6: Graph of focal length for lens D
Table 21: Data presentation for Convex lens E
(cm)^{-1} |
(cm)^{-1} |
(cm)^{-1} |
0.07 |
0.04 |
0.11 |
0.05 |
0.05 |
0.10 |
0.04 |
0.06 |
0.10 |
0.03 |
0.07 |
0.10 |
0.03 |
0.07 |
0.10 |
0.03 |
0.08 |
0.10 |
0.02 |
0.08 |
0.10 |
Figure 7: Graph of focal length for lens E
- PART B
Table 22: Data presentation for Trial 1
(cm)^{-1} |
(cm)^{-1} |
(cm)^{-1} |
0.03 |
0.02 |
0.05 |
0.03 |
0.03 |
0.05 |
0.02 |
0.03 |
0.05 |
0.02 |
0.03 |
0.05 |
Figure 8: Graph of focal length for Trial 1
Table 23: Data presentation for Trial 2
(cm)^{-1} |
(cm)^{-1} |
(cm)^{-1} |
0.03 |
0.02 |
0.05 |
0.03 |
0.03 |
0.05 |
0.02 |
0.03 |
0.05 |
0.02 |
0.04 |
0.05 |
Figure 9: Graph of focal length for Trial 2
Table 24: Data presentation for Trial 3
(cm)^{-1} |
(cm)^{-1} |
(cm)^{-1} |
0.04 |
0.02 |
0.05 |
0.03 |
0.03 |
0.05 |
0.02 |
0.03 |
0.05 |
0.02 |
0.04 |
0.05 |
Figure 10: Graph of focal length for Trial 3
Analysis: My expected graph of against would be a linear one and the focal length would be the y intercept as well as the x- intercept. This is because if we insert 0 in the place of or, in the formula , then it equals the value of . All the above graphs are linear, straight- line graphs that have their y-intercepts as the value of the focal length and that it matches the values of <img src="https:/
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