Determination of the Focal Length of a Convex Lens
Info: 4987 words (20 pages) Essay
Published: 9th Mar 2018 in
Physics
 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 67 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 18 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 15, 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) 
(fx) 
(fx)^{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) 
(fx) 
(fx)^{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) 
(fx) 
(fx)^{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) 
(fx) 
(fx)^{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) 
(fx) 
(fx)^{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) 
(fx) 
(fx)^{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) 
(fx) 
(fx)^{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) 
(fx) 
(fx)^{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
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