Focal Length of Lenses

Objective

The objective of this experiment was to find the focal length of two converging lenses separately and also to find the focal length of combination of two converging lenses. Another objective was to study the image formed by a converging lens plus diverging lens.

Background

This experiment focused on a simple lens system. A simple lens is a piece of glassware that is spherical on one side and flat on another side, or spherical on both sides. There are two different types of lenses. A converging lens, also called a positive lens, and a diverging lens, also known as a negative lens. A converging lens thicker middle area than top and bottom, and the diverging lens has a skinnier mid-section.

When a parallel ray passes through the lens, the rays will be diffracted so that it will always intercept the focal point. When a ray passes through the optical center, there will be no diffraction. The three basic rules about the rays are:

1) Rays travelling parallel to the lens will diffract towards the focal point

2) Rays travelling at optical centre will have no diffraction.

3) Rays travelling through the focal point towards the lens will have a parallel diffraction to the plane.

Focal length f varies from lens to lens and it is characterized by:

1/f = 1/do + 1/di (equation 1)

do represents the distance of the object to the lens and di represents the distance of the image formed from the lens. Rearranging for f, the equation becomes:

f = dido / (dÐ'¬i + do) (equation 2)

The theory behind the experiment Part I is that when an object that is extremely far away, the focal length is equal to di. Using equation 1, this idea is supported. When 1 is divided by do (infinity), the value will get closer to zero, so it’s possible to assume the limit as zero. If this idea holds, the new equation will become 1/f = 1/di. Thus f = di. Using this theory, the focal length of converging lens A was calculated. Also to support the theory, the same converging lens’s focal point was calculated using a finite object. Then the percentage difference was calculated using the equation:

%difference = |(fwindow - fexperimental)| / fwindow * 100% (equation 3)

The focal length for the experiment was gathered using a linear graph. Equation 1 could be rearranged into a linear equation.

1/do = 1/f + 1/di

1/do = - do/dido + 1/f (equation 4)

The slope would be вЂ"do/di and the y-intercept would be 1/f according to this equation.

For Part II and III, the equation that relates the focal length of two lenses to create a compound lens system was used.

1/fc = 1/fÐ'¬1 + 1/f2 (equation 5)

Isolating this equation for f1, the new equation becomes:

f1 = fcf2 / (fÐ'¬2 - fc) (equation 6)

Using compound lens experiment and understanding the theory behind equation 5, it was possible to calculate the focal length of a compound lens system by measuring the focal length of individual lenses. It is also possible to measure the focal length of a single lens and a compound lens system then find the focal length of the unknown lens. This idea works for both converging and diverging lenses.

Procedure

Part A: Single Converging Lens

Fig 1 Lens experiment setup

One convergent lens was placed between the light source and the white screen on the optical rail. Either lens or the screen or both were moved to focus a real image of the crossed arrows on the screen. Then the distance of the object to the lens (do) and the image distance to the lens (di) were measured and recorded the image orientation. To measure the object size and the image size, a ruler was used and linear magnification was calculated. Same procedure was repeated for five different object distances. The focal length was determined by the y-intercept points of the best fit line by plotting graph of 1/di vs. 1/do.

Part C: Compound Lens (Two Converging Lenses)

Same steps from Part A were followed to find focal length f2 of a second converging lens. Then two lenses were placed side by side, as closely as possible and then the focal length of the compound lens fc was determined. The relationship between 1/fc =1/f1+1/f2 was used to determine the theoretical value of fc. A comparison was made between the theoretical value of fc and the value that was estimated to determine which method was more accurate.

Part E: Compound Lens (Convergent + Divergent Lens)

One converging lens and one diverging lens were placed side by side, as closely as possible and used to estimate the focal length of the compound lens. Same formula as Part C was used to find the focal length fc of the diverging lens. Using the results an explanation was given of why convergent lenses are also called “positive” while diverging lenses are also called “negative.”

Observation Data

Part I

Table 1 Distance of window and its image from a converging lens A

do(cm) di(cm)

в?ћ 11

Table 2 Distance of an object and its image using a converging lens A

Trial do (cm) di (cm) ho (cm) hi (cm)

1 19 30.0 1 2.0

2 17 36.0 1 2.3

3 21 26.5 1 1.5

Part II

Table 3 Distance of an object and its image using a converging lens B

Trial do (cm) di (cm)

1 18.5 30.0

2 20.0 27.5

3 18.0 34.0

Table 4 Distance of an