Shaikh Ilyas
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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]
[2]
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:
[3]
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 ?AHO,
In ?EDO,
?
—– (1)
In ?BOC,
In ?EDC,
?
—— (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: ?m = = = 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: ?m = = = 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: ?m = = = 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: ?m = = = 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: ?m = = = 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: ?m = = = 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: ?m = = = 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: ?m = = = 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
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