### Purpose

To develop an understanding of how light is refracted, to apply the concept of refraction to the study of how thin lenses form images, to learn to draw ray diagrams to assist in the predicting the locations of images formed by spherical thin lenses, and to determine the focal length of a converging lens using three different methods.

### Hypothesis

When light hits the convex lens, it will refract and converge at a point in front of lens that will be construed from the rays diverged behind the lens. The distance between this point of convergence and the lens is the focal length, which should be similar in value to an indirect measurement of the focal length of the converging lens by varying the distance between the lens and an object. Using the distance between the lens and object and the lens and projected image, the focal length will be able to be determined using the thin lens equation 1/p + 1/q = 1/f. As the object is moved further past the curvature, the image will become smaller. The closer the object is moved to the lens between the curvature and focal length, the image will become larger. While the object is beyond the focal length, the image will be real and inverted. Once the object is placed between the focal length and lens, the image will become virtual and upright. Finally, the focal length interpreted using the conjugate method should also be similar to the previous focal length measurements.

### Labeled Diagrams

See attached sheet.

### Data

Part 1:

Position of object | Real or virtual image | Upright or inverted image | Image smaller or larger than object | Location of image |

1. Beyond C | Real | Inverted | Smaller | Between F and C |

2. At C | Real | Inverted | Normal | At C |

3. Between C and F | Real | Inverted | Larger | Beyond C |

4. At F | No image | No image | No image | No image |

5. Between F and the lens | Virtual | Upright | Larger | Between F and C |

Part 2A:

f = 7.70 cm

Part 2B:

p (cm) |
q (cm) |
f(cm) |

20.0 |
11.8 |
7.42 |

17.0 |
13.2 |
7.43 |

15.4 |
14.4 |
7.44 |

13.0 |
17.3 |
7.42 |

11.5 |
19.7 |
7.26 |

f_{average} = 7.40 cm

Part 2C:

First conjugate position (cm) |
Second conjugate position (cm) |

43.3 |
10.2 |

f = 7.41 cm

### Questions

1. What is the measured value of the focal length and how does it compare with the given value of focal length for the length for the lens?

The measured focal length is 7.70 cm which is fairly close to the given value of the focal length, 7.50 cm. The percent difference is 2.63%.

2. For each p and q in your data table above, calculate f. Record in your data table.

See data table.

3. Calculate and record the average value of your f’s.

The average value of the focal lengths is 7.40 cm.

4. How does the average focal length compare with the focal length printed on the lens? What is the percent difference?

The average focal length calculated is slightly less than the focal length printed on the lens. The percent difference is 1.34%.

5. Calculate the focal length of the lens using f = (D^{2} – d^{2}) / 4D where D is the distance between the object light and the image screen and d is the distance between the two conjugate positions.

D = 51.1 cm and d = 33.1 cm, so f = 7.41 cm.

6. How does this value for the focal length compare to the given value of the focal length for the lens? What is the percent difference?

This value for the focal length is again slightly less than the given focal length of the lens. The percent difference is 1.21%.

7. How do your focal lengths calculated from Parts 2A, 2B, and 2C compare? What is the percent difference between f from 2A to 2B? 2A to 2C? 2B and 2C?

The focal lengths from part 2B and 2C are nearly identical, while 2A is a tad larger than each of them. The percent difference between f from 2A to 2B is 3.97%, from 2A to 2C is 3.84%, and from 2B and 2C is 0.135%.

### Conclusion

During part 1 of the experiment, ray diagrams of thin lenses were constructed for the following scenarios: an object beyond the curvature, an object at the curvature, an object between the curvature and focal length, an object at the focal length, and an object between the focal length and lens itself. The image size, orientation, location, and realness were predicted and recorded for each scenario. These predictions were used to aid with part 2B of the experiment.

During the half of the experiment, the focal length of a positive lens was measured in three different manners. The first of these methods, part 2A, used the application of a direct measurement to find the focal length. Equipment was set up on the optics bench so that light shone through a parallel ray lens, slit plate, and 75 mm convex lens onto a ray table. The parallel ray lens was first adjusted in order to align the light rays in a parallel manner before placing the 75 mm convex lens in front of the ray table. The light rays refracted through the lens and diverged onto the ray table. The outermost rays were traced on a piece of paper and were extended to meet at a focal point. The distance between this focal point and the lens was measured as 7.70 cm and recorded as the focal length. This was 2.63% difference from the given value of 7.50 cm as the focal length of the lens. This error can be attributed in part to difficulty keeping the tracing paper in place. It undoubtedly shifted some during tracing of the light rays. It was also difficult decipher the exact distance from that drawn point to the lens, as they were not on the same plane; the ray table was elevated and at an angle in relation to the lens. These attributions are most likely what led to that slight error.

During part 2B, the thin lens equation, 1/p + 1/q = 1/f, was used to measure the focal length. The crossed arrow target was placed at five positions beyond the measured focal length from of the lens part 2A. The distance from the lens to object (crossed arrow target) and from the lens to focused image were recorded for each instance. As the object was moved further away from the lens, the image became smaller, and as it was moved closer to the lens, the image became larger. The image was real and inverted in all cases. The average focal length from these recordings was calculated to be 7.40 cm, which was 1.34% different than the given focal length of the lens. The minor error could be attributed to difficulty locating the exact position where the image was truly focused; there was not definitive way to tell if the image was focused or not. There was likely some error in the interpretation of the positioning of the instruments as well, due to their thickness. Each measurement may have been misread by a couple tenths of a centimeter.

During part 2C, the conjugate method was used to measure the focal length. The method called for the crossed arrow target to be placed directly in front of the light source with the viewing screen at the opposite end of the optics bench, as far away as possible. The 75 mm convex lens was positioned between the object and viewing screen, adjacent to the crossed arrow target. The lens was slowly moved towards the viewing screen until a large focused image came into picture on the viewing screen. This positioning was noted and recorded as the first conjugate position. The lens was then moved towards the viewing screen until a minute focused image came into picture. This positioning was noted and recorded as the second conjugate position. The distance between these two positions was recorded as d and the distance between the crossed arrow target and viewing screen was recorded as D. These values were subbed into the equation f = (D^{2} – d^{2}) / 4D to find the focal length. It was calculated to be 7.41 cm using this method, which is 1.21% different than the given focal length. Error from this part of the experiment could again be attributed to possibly not getting the image in perfect focus, which would through off the positioning. Moreover, it was also difficult to read the exact positioning of the lens, object, and image as these apparatuses all had a thickness to them. As in part 2B, there may have been a couple tenths of a centimeter error in each reading. The measured and calculated focal lengths from all three methods were fairly consistent. The focal lengths from part 2B and 2C were nearly identical, while 2A was a small amount larger than each of them. The percent difference between f from 2A to 2B was 3.97%, from 2A to 2C was 3.84%, and from 2B and 2C was only 0.135%.

### Equations

C = 2f

1/p + 1/q = 1/f

f = (D^{2} – d^{2}) / 4D

Percent Difference = |x_{1} – x_{2}| / (x_{1} + x_{2})/2 x 100%