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FOCAL LENGTH OF LENSES INTRODUCTION A simple lens is a piece of glass or plastic having two polished surfaces that each form part of a sphere. One of the surfaces must be curved and the other surface may be curved or flat. An example of a simple lens would be obtained if a piece of a glass ball were sliced off as shown in Figure 1 below. Lens

Glass Ball

Figure 1. Lens obtained by slicing off a piece of a glass sphere. The piece of the ball sliced off would be a lens with a spherical side and a flat side. Lenses can be made in a variety of shapes for various applications. Some examples of lens shapes are illustrated below.

Figure 2. Different types of convergent and divergent lenses A lens thicker in the center than at the edge is called a converging or positive lens whereas a lens thinner at the center than at the edge is called a diverging or negative lens. In the illustration shown, lenses 1, 2, and 3 are converging or positive lenses. Lenses 4 and 5 are diverging or negative lenses. With thin lenses (the thickness at the center of the lens is not too great) a mathematical approximation can be used. This approximation assumes the bending of light occurs in one plane inside the lens. A ray of light coming from a very distant object, such that the ray is parallel to the optical axis (Ray #1 or Ray #2), will be bent by refraction at the two surfaces of the lens and will cross the optical axis (PA) at the focal point F2 (and F1 respectively) of the lens, as in the figure below. F1 and F2 are symmetrical with respect to the optical centre O.

Ray #1 Ray #2

PA

F1

O

F2

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Figure 3. Ray #1 and Ray #2 enter the lens parallel to the PO A ray passing through the optical center O of the lens will emerge from the lens undeviated. . PA

O

Figure 4. A ray of light passing undeviated through the optical centre O The image of a real object formed by a convergent lens was graphically reproduced in a ray diagram below, using the following rules: • Ray 1: The ray leaving the tip of the object and traveling parallel to the principal optical axis (PA), will pass through the focal point F2 after passing through the lens. • Ray 2: The ray leaving the tip of the object and passing through the focal point F1 will emerge from the lens traveling parallel to the principal optical axis (PA). • Ray 3: The ray leaving the tip of the object and passing through the center (O) of the lens will emerge from the lens undeflected.

ho F2 F1

hi f

f

2f

2f do

di

Figure 3. Ray diagram showing the image formed by a convergent lens. Its characteristics are: real, inverted, smaller than the object The focal length, f of a lens is a unique characteristic of the lens that may be defined as the distance from the lens at which a distant object will produce an image. Focal length is

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related to the object distance (do ) and the image distance (di ) by the following thin lens equation: 1 1 1 = + (1) f do di Note: The assumption that the lenses are “thin” does not introduce a significant error!

OBJECTIVES 1. In this experiment you will analyze the properties of convergent and divergent lenses and determine their focal length and magnification. 2. You will also analyze a simple device made up of two convergent lenses: the telescope. MATERIALS • Optical bench with measuring tape attached • Lens holders that clip directly to track, with position indicators • Convergent and divergent lenses mounted in lens holders • White screen, • Light source with crossed arrows object • Ruler. PROCEDURE PART A: SINGLE CONVERGING LENS 1. Place one convergent lens between the light source and the white screen on the optical rail. Move either the lens or the screen or both to focus a real image of the crossed arrows on the screen. 2. Measure on the optical rail the object distance to the lens (do) and the image distance to the lens (di). 3. Record the image orientation. Measure with a ruler the object size (ho) and the image size (hi) and calculate the linear magnification: h M= i h0 Table 1. Single Converging Lens 1 1 do di ho hi Image di d0 No (cm) (cm) (cm) (cm) Orientation M=hi/ho di/do (cm-1) (cm-1) 1. 2. 3. 4. 5.

4. Repeat for a total of 5 distinct object distances. Determine the focal length of the lens by making a plot of 1/di vs. 1/do . The focal length of the lens can be determined from the y-intercept of the best fit line (see Figure 4 on the next page).

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1 (m −1 ) do

1 f

1 (m −1 ) di

1 1 1 equals vs. f di d0

Figure 4. The y-intercept of the graph of

5. Compare the values of the linear magnification with those in the last two column and try to interpret.

PART B: THE MAGNIFIER A converging lens can sometimes be used as a magnifying glass. When used this way you are looking through it at a virtual, enlarged image. The object has to be placed at a distance less than the focal distance from the optical centre of the lens (see Figure 5).

O

F IMAGE: • VIRTUAL • UPRIGHT • ENLARGED

F

OBJECT: REAL f

f

Figure 5. Image formed with a magnifying glass

1. Use the lens in part A as a magnifier and examine the characteristics of the image. 2. Are these characteristics changing, assuming that the position of the object remains between F and O?

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PART C: COMPOUND LENS (TWO CONVERGING LENSES) 1. Get a second converging lens and estimate its focal length f2, by repeating, steps 1 to 4 in part A. Table 2: Second Converging Lens 1 d di o d0 No (cm) (cm) (cm-1)

1 di (cm-1)

1 f2 (cm-1)

f2 (cm)

1. 2. 3. 4. 5.

2. Position now the two lenses side by side, as close together as possible, and determine the focal length of the compound lens fc, proceeding as before. Table 3: Compound Lens (Both Converging Lenses) 1 1 d d o i d0 di No (cm) (cm) -1 (cm ) (cm-1)

1 fc (cm-1)

fc (cm)

1. 2. 3. 4. 5.

3. Now, using the relationship: 1 1 1 = + fc f1 f 2 determine the theoretical value of fc. 4. Compare the value of fc with the value found in step 2. 5. Which method appears to be more accurate? Explain why.

(2)

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PART D: SINGLE DIVERGING LENS 1. Position a diverging lens anywhere on the optical rail between the light source and the white screen and try to obtain a real image. Draw a ray diagram and explain your conclusion. 2. What are the characteristics of the only type of image you can obtain with a diverging lens? Draw a ray diagram and explain. PART E: COMPOUND LENS (CONVERGENT + DIVERGENT LENS) 1. Position now one of the converging lenses used before and the diverging lens side by side, as close together as possible, and estimate the focal length of the compound lens proceeding as before. Make sure the combination produces a converging system. Table 4: Single Diverging Lens

No

do (cm)

1 d0 (cm-1)

di (cm)

1 di (cm-1)

1 fc (cm-1)

Average 1 fc (cm-1)

fc (cm)

1. 2. 3. 4. 5.

2. Using formula (2) for the focal length fc of a compound lens calculate the focal length of the diverging lens. 3. Could you explain now, why convergent lenses are also called “positive” while diverging lenses are also called “negative”? PART F: THE TELESCOPE A telescope is a device that has the ability to make distant objects appear much closer and brighter. A telescope with an objective lens is called a refracting telescope or a refractor. A telescope with an objective mirror is called a reflector. The refracting telescope contains an objective, a lens that has the function to collect the light from distant objects and bring it to a focus very close to its focal point. The image thus formed is real, inverted and naturally smaller than the real object. Therefore, the role of the objective is not to produce a larger image, but to bring the real image of the object closer. The second lens of the telescope, the eyepiece, is a smaller lens which works as a magnifying glass to enlarge the image created by the objective. The magnification of a telescope is defined as the ratio of the focal length of the objective lens to the focal length of the eyepiece: Magnification =

f objective f eyepiece

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The brightness of the image depends on the aperture (size of the diameter) of the objective. The telescopes can be classified according to the pair of values for the aperture and focal length pf the objective: Long focal length telescopes optimized for high magnification planetary observing typically perform worse on deep sky objects than short focal length low magnification scopes. Your scope will meet your needs and desires only if you consider each of these design parameters and compare them to your planned uses. Better yet, build two or three telescopes, each meeting a different design goal. Then, you really can have the best of all worlds! A diagram of a simple refracting telescope like the one that you are to construct is shown below.

Objective

Eyepiece (Ocular)

Eye

FObjective

FEyepiece Parallel Incoming Rays Final Image

Figure 6. Refracting Telescope (Not to scale!)

The distant object is to the right of the diagram. Notice that the distance between the two lenses is very close to the sum of their individual focal lengths. To construct a refracting telescope, hold the two converging lenses and point them toward a distant object in the room. Make sure the lens with the largest focal length is the objective and the lens with the shortest focal length is the eyepiece. Align the two lenses so that a straight line runs between the object and the two lenses. 1. Record the focal length of the lenses used in your telescope. 2. Look through the eyepiece of your telescope and adjust the length between the two lenses to produce the sharpest image. Is the image upright or inverted? Is the image larger or smaller than the object? 3. Compute the magnification of your telescope. Does the result of this calculation correlate with your observation? Record the results of your computations and observations in your lab notebook. fo (cm)

No 1. 2.

fe (cm)

M=fo/fe

Final Image Characteristics