Images with “Thin” Lenses

Updated 2/2017 jmh, rc

Materials

1 each PASCO Basic Optics Light Source

2 each lenses: one with a negative focal length, the other with a positive 200 mm focal length

1 each empty lens holder

1 each optical track (optical bench)

1 each optical target screen

1 each metric ruler

Goals

1) To understand the nature of image formation from thin lenses.

2) To learn experimental techniques for determining the focal lengths of positive (converging/convex) and negative (diverging/concave) lenses in conjunction with the thin-lens equation.

3) To learn how to make a scale “ray diagram” for a combination of a positive and negative lens using three principle rays for each lens and interpret it.

4) To understand the specific meaning of the term, “magnification,” as applied to optical systems and to determine its value by three methods: a) direct measurement, b) calculation using the thin lens equation, and c) using the ray diagram.

Introduction

Definitions

Object distance do: This is the distance from a lens to the object being viewed. This distance is always positive except in rare cases involving more than one lens. In this lab we will only ever deal with positive object distances.

Image distance di: This is the distance from a lens to the image it has formed. This distance is considered positive if it is on the opposite side of the lens from the object. It is considered negative if the image is on the same side of the lens as the object.

For any simple focusing element it can be shown that

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

where do and di are the object and image distances respectively and f is the focal length of the element. This is called the thin-lens equation.

Figure 1: Ray diagram illustrating principal rays.

The Apparatus

An optical bench with a metric length scale attached to it, two lenses and holders, a light source light source, and a viewing screen are provided. Use the end of the light source with the crossed arrows as the object to be imaged. Please do not touch the glass lenses.

Exercise 1: Qualitative Investigation of Image Formation

Understanding image formation:

Predict what will happen to a real image formed by a converging (convex) lens when you cover up the top half of the lens. Record these predictions and have your instructor initial them before continuing. Predictions that have not been initialed will not be graded. Neither will the accompanying explanations. Predictions will not be initialed after you have set up your lens and light source.

Place the 200mm focal length converging (convex) lens on the optical track, near the center. Place the light source at one end of the track and the viewing screen at the other. Adjust the lens and screen locations until you can see a real image in focus on the viewing screen. Cover up the top half of the lens and observe any changes that take place in the image as the lens becomes blocked. Explain this in your report.

Determining the Focal Length of a Positive (converging/convex) Lens

We will be using the optical bench with the light source and the viewing screen to determine the focal lengths of the two lenses provided, one positive (converging/convex) and one negative (diverging/concave). You should be able to tell which is which by looking at the cross-sections.

For now you will be determining the focal length of the (nominally 200 mm) positive lens. This can be done experimentally by finding pairs (find several quite different values) of object and image distances that give clear real images (a “real” image can be projected on a screen) of the light source on the viewing screen. Determine your uncertainty in image distance by shifting the lens until the image blurs noticeably. Then use the thin lens equation to calculate the focal length. Repeat this five times using significantly different values of do and di (you may assume the same uncertainty for each measurement). Find the mean value of the focal length and compute its standard deviation. Does your value for focal length (with uncertainties) match with the actual focal length found printed on the lens holder?

Locating Virtual Images by Parallax:

You can determine the relative distance of different objects using a simple technique called parallax. Pick a distant object (maybe a fire extinguisher, or a chair across the room.) Hold up a pencil or calculator as a nearby reference object. Cover one eye and observe both objects while moving your head back and forth slightly. (If anyone makes fun of you, tell them it is for science, and immediately report them to your instructor.) The object which moves further across your field of view must be closer to you. As an extreme example, this is why the moon seems to hang in a single spot in the sky even as you drive all the way across town.

Remove the screen and bring the +200mm focal length lens to the end of the track at 0 cm. Make a simple object by taping a string vertically across the empty lens holder. Place the lens holder so that the string is 15 cm from the lens. This should cause the lens to act as a magnifying glass, forming a magnified virtual image behind the lens.

You can use the parallax technique to (roughly) locate the virtual image formed by your “magnifying glass”. View the image through the lens with one eye covered. Have a partner hold a tall reference object (a meter stick works well) at different locations directly behind the lens (adjust this position in 5 cm increments). When moving your head back and forth causes the magnified string image to move in synch with the meter stick, you will know they are both the same distance away. Determine the image distance uncertainty by determining how far the ruler can move without you noticing the loss of parallax alignment with the image. Note: It is important that you watch the meter stick move outside the lens while watching the image move inside the lens. You may be able to see the meter stick through the lens as well, but you want to reference the real meter stick and its actual distance from your eye.

Measure the (negative) image distance for one object distance with this set up, and compare it to the one you can predict using the thin lens equation. Make sure to include your measurement uncertainties as discussed in lab based on the range of distances where the parallax motion appears to align.

Exercise 2: Determining the Focal Length of a Negative Lens

Set the light source, concave lens, and screen on the track. Can you produce an image on the screen with the concave lens using this setup?

Using both lenses (place the negative lens nearest the light source) again find locations of the lenses and screen that result in a clear image. Note that the image should be large enough on the screen to tell if the image in focus. Place the light source at xo = 0.0 cm. Record the positions of the concave lens (x-), convex lens (x+), and screen (xs) for each of the five trials. At each measurement, also record the diameter of the real image circle (yi) projected on the screen. This will be used to test the magnification equation in Exercise 4. The object circle has a diameter of yo = 4.00 cm.

With this configuration we only can measure the object distance of the negative lens, and the image distance of the positive lens. See Figure 2.

Figure 2: Sample ray diagram illustrating image formation by a combination of negative and positive lenses.

Knowing the focal length of the positive lens from Exercise 2, the thin lens equation can be used to find the object location for only that lens. In this case, the object for the positive lens is the image, created by the negative lens. Once you have found this point on your optical bench/track, you can measure from there to the negative lens, to find the image distance for the negative lens.

You will now have both the object distance (measured from lamp) and image distance (measured from calculated location of positive image object). Apply the thin lens equation again to find the focal length of the negative lens. Note that the sign conventions used demand that the focal length for a diverging lens is designated as a negative number. Repeat this process for five significantly different locations of the lenses and viewing screen. Create an Excel file to do these calculations. Have your instructor check your calculations. You must show one sample calculation in your report.Calculate the mean focal length and the corresponding standard deviation and compare to the actual focal length found on the lens holder.

Exercise 3: A Ray Diagram (Complete at home):

Pick one configuration of lenses and viewing screen distance from Exercise 2. Choose a configuration where the calculated value for the negative lens is closest to the nominal value (typically -150 mm). Draw a complete ray diagram to scale (not 1:1, make it fit on one sheet of paper) showing the formation of the intermediate image from the negative lens and the final image of the positive lens. Use the experimentally determined focal lengths for the lenses from the configuration you are using to create the ray diagram. Use different horizontal and vertical scales to exaggerate the object height relative to the length of the optical track. Utilize a straight edge to draw ray paths. Assume ray paths bend only at the center of each lens. Trace the rays for the lens closest to the light source first; then use the resulting image as the object for the second lens. How well does your ray diagram predict the correct location for the final image? Compare to your experimental value from Ex. 2.

Exercise 4: Magnification (Can be completed at home if all data was collected in Ex.2)

The magnification (m) is defined as the ratio of the image size (yi) to the size of the object (yo) being imaged (m = yi/yo). When the image is upside down, the magnification is negative. If the image is upright, that is the same orientation as the object, the magnification is positive. From the ray diagram for either a positive or negative lens the magnification (sometimes called the transverse magnification), m, can be shown to be equal to m = di/do. For the two-lens system, the virtual image of the negative lens has magnification m- = di-/do-and this virtual image is magnified by the positive lens m+ = di+/do+, so the total magnification is m- m+ = (di-/do-)( di+/do+).

For the same configuration used in Exercise 3, calculate the magnification of the two-lens combination as shown on the viewing screen (m = yi/yo), and compare this value to that calculated with the aid of the thin-lens equation: (m- m+ = (di-/do-)(di+/do+)). You should also be able to determine the magnification directly from your ray diagram in Exercise 3 using the equation (m = yi/yo). Compare the three magnifications you calculated in this exercise using % Differences.

Before you leave the lab

Straighten up your lab station.

Report any problems or suggest improvements to your lab instructor.