The purpose of this lab experiment is apply the thin-lens equation and the mirror equation to a few real-world optics applications.

In class you have studied the physics of thin lenses and spherical mirrors. In today's lab, we will analyze several physical configurations using both biconvex lenses and concave mirrors. The components of the experiment, that is, the optics device (lens or mirror), object and image screen, will be placed on a meter stick and may be repositioned easily. The meter stick is used to determine the position of each component. For our object, we will make use of a light source with some distinguishing marking, such as an arrow or visible filament. Light from the object passes through the lens and the resulting image is focused onto a white screen.

One characteristic feature of all thin lenses and concave mirrors is the focal length, , and is defined as the image distance of an object that is positioned infinitely far way. The focal lengths of a biconvex lens and a concave mirror are shown in Figures 1 and 2, respectively. Notice the incoming light rays from the object are parallel, indicating the object is very far away. The point, , in Figure 2 marks the center of curvature of the mirror. The distance from to any point on the mirror is known as the radius of curvature, . It can be shown that is twice the focal length.

Figure 1. The focal length of a biconvex lens.		Figure 2. The focal length, radius of curvature and center of curvature of a concave mirror.

Thin Lenses

A common experimental setup for a lens experiment is shown in Figure 3.

Figure 3. The lens experimental setup consists of a light source (object), converging lens and image screen. These components are placed on a meter stick for easy position measurements. Notice the image is inverted. Click on the image to enlarge it.

When the object is outside the converging lens' focal point, , the resulting image is real, inverted and on the side of the lens opposite the object. This is shown with the geometrical ray diagram of Figure 4.

Figure 4. An object outside the lens' focal point forms a real and inverted image on the side of the lens opposite the object.

The above figure shows the object distance, , and the image distance, . Each of these distances are measured from the center of the lens. In addition, the object height, , and the image height, , are also shown.

The parameters , and , are related by the thin lens equation which is given by

(1)

The magnification of the lens, , is defined as the ratio of the image height, , to the object height, , or

(2)

For the thin lens, the magnification is also equivalent to the negative ratio of the image distance to the object distance, or

(3)

A positive value for in Equation 3 indicates that the image is upright and on the same side of the lens as the object. A negative means the image is inverted and appears on the opposite side of the lens as the object.

The situation is very different, however, when the object is between the focal point and the lens. As shown in Figure 5, this configuration creates a virtual image on the same side of the lens as the object, which is upright and larger than the object.

Figure 5. An object inside the lens' focal point forms a virtual and upright image. The image is always larger than the object and appears on the same side of the lens as the object.

The Refracting Telescope

A refracting telescope uses a combination of lenses to form a magnified image. For example, the telescope shown in Figure 6 incorporates two lenses, an objective lens and an eyepiece lens, for this purpose. The eyepiece lens has a focal length , and the objective lens has a focal length, . To focus the telescope, the eyepiece is positioned so that the focal point of both lenses are coincident. In other words, the distance between the lenses is .

Figure 6. The lens configuration of the refracting telescope. The image will be in focus if the focal point of both lenses coincide.

It can be shown that the angular magnification of the telescope can be given either in terms of the apparent size of the image and object, or in terms of the focal lengths of the lenses:

(4)

The negative sign in Equation 4 indicates that the image is inverted.

One way to measure the apparent size of an object positioned far away from you is to use a vernier caliper as shown in the figures below. If we wish to determine the length of the arrow on the blackboard, first focus your eyes on the arrow and adjust the jaws of the caliper around the image. Then focus your eyes onto the caliper scale and take a reading, in this case, 2.68cm. This technique can be repeated when viewing the arrow through a telescope. The important thing to remember is that the vernier caliper must remain a constant distance from your eye if you are to compare apparent sizes with and without the aid of the telescope.

Figure 7. Click on images to enlarge them.

Convex Mirrors

Before reading this section, refer back to Figure 2 for a graphical description of the mirror parameters. A common experimental setup for a mirror experiment is shown in Figure 8.

Figure 8. The mirror experimental setup consists of a light source (object), convex lens and image screen. The mirror and light source are placed on a meter stick for easy position measurements. The back of the mirror is shown in the foreground and the image of the filament is projected onto the white card. Click on the image to enlarge it.

When the object is outside the concave mirror's radius of curvature, , the resulting image is real, inverted, smaller than the object and on the same side of the mirror as the object. This is shown with the geometrical ray diagram of Figure 9.

Figure 9. When an object is placed outside the mirror's center of curvature (point C) the image that is formed is real, inverted and is smaller than the object.

The above figure shows the object distance, , and the image distance, , of an object placed outside the mirror's center of curvature, . Each of these distances are measured from the mirror's center (point V). The parameters , and , are related by the mirror equation, which is identical to the thin lens equation (Equation 1),

(5)

Additionally, the mirror equation may be written in terms of the mirror's radius of curvature,

(6)

The magnification of the mirror is determined exactly as we did with lenses and is given by Equations 2 and 3.

The Lens Objectives
1. Use a meter stick and white screen to quickly estimate the focal lengths, of both lenses to the nearest five centimeters. Note, it is not necessary to use the optics bench for this.

2. Setup the lens apparatus as shown in Figure 3, using the lens with the longer length. Record , , and for many different relative positions of the object, lens and image screen.

3. Using data from Objective 2, make a plot of versus and answer the following questions:
   1. What is the relationship between and ?
   2. As the object distance, , becomes large, what approximate value does approach? Physically, what does this value represent? Can you compare this value to a measured quantity to ascertain if you are correct? Can you verify this using Equation 1?
   3. Using the graph, determine the range of positions for the object that will produce virtual images. Can you verify this using the equipment?

4. Make a plot of versus and determine the value of the lens' focal length, .

5. Make a plot of versus and determine the value of the lens' focal length, .

6. For each data point taken in Objective 2, calculate the magnification of the object size using Equation 2. Also calculate using Equation 3 and compare your results for each data point.

   The Telescope Objective

7. Use both of your lenses to construct the simple telescope shown in Figure 15. Your TA will draw an image on the black board and you will use your telescope and a vernier caliper to verify Equation 4. Refer to the background section on the refracting telescope to assist you with your measurements.

   The Mirror Objectives

8. Use a meter stick and white screen to quickly estimate the focal length, of the concave mirror to the nearest ten centimeters. Note, it is not necessary to use the optics bench for this.

9. Setup the mirror apparatus as shown in Figure 8. Record and for many different relative positions of the object, mirror and image screen. Use this data to determine the focal length, , and the radius of curvature, of the concave mirror.

(Figure 10.) The light source has an image of an arrow drawn on it to represent the object. Note the location of the position indicator in relation to the object position. (Figure 11.) You will use two lenses for this experiment. Note that one lens has a longer focal length than the other. (Figure 12.) The image screen. The image is projected onto the screen. (Figure 13.) The meter stick and support stands. (Figure 14.) The vernier caliper is used to measure image and object sizes. (Figure 15.) A simple telescope using objective and eyepiece lenses. (Figure 16.) Objectives 8 & 9 require the use of a spherical concave mirror like the one pictured here. (Figure 17.) For Objective 9 you will need to replace the orange bulb with this cleam one, since you are asked to focus on the filament of this light bulb. (Figure 18.) The box of concave mirrors.

[Click on images to enlarge.] 10 11 12 13 14 15 16 17 18

1. Caution!!! Handle the equipment with care. The lenses and bulbs are made of glass and obviously will break if handled roughly.

2. Caution!!! Plug the light source into 120VAC power outlets. (117VAC is okay, too!) When in doubt, use the white adapters with six outlets.

1. Clemson Physics Lab Tutorials
2. Measurement uncertainties
3. Using Excel
4. Using error bars in Excel

Each lab group should download the Lab Report Template and fill in the relevant information as you perform the experiment. Each person in the group should print-out the Questions section and answer them individually. Since each lab group will turn in an electronic copy of the lab report, be sure to rename the lab report template file. The naming convention is as follows:

For example the group at lab table #5 working on the Ideal Gas Law experiment would rename their template file as "5 Gas Law.doc".

These Nudge Questions are to be answered by your group and checked by your TA as you do the lab. They should be answered in your lab notebook.

General Nudges

1. What do the variables , , , , and mean physically. How can you measure each?
2. From what point on the biconvex lens are , , and measured?
3. From what point on the concave mirror are , , and measured?
4. Is it possible for virtual images to be projected onto the image screen?

1. Which lens are you using -- the thinner lens or the thicker lens?
2. What method will you use to determine the focal lengths?
3. What will you use for your object? Is your object near to you or far away? Does it matter?
4. What was your estimate of the lenses' focal lengths?
5. What is the uncertainty in your measurement?

1. How will you measure the positions of the object, lens and image?
2. How will you measure , , and ?
3. What is the uncertainty of these measurements?
4. Does it matter which electrical socket the bulb is plugged into on the bulb holder?
5. If the image screen is not perpendicular to the meter stick, how will your data be affected?
6. If the object and image positions are fixed, how many real images can you create simply by moving the lens?
7. Data taken in Objective 2 will be used for Objectives 3 - 6. Objective 3 is especially sensitive to the number of data points. Looking ahead to Objective 3, what values of and are most critical? Why?

1. If you were unable to answer the questions, perhaps you should review the nudge questions for Objective 2.

1. How will you calculate ?
2. What was your value of the lens' focal length?
3. What value for the slope do you expect? Is this what your data shows?
4. What are the magnitude of error bars for the data points?
5. What is the uncertainty in your measurement of ?

1. Why did we decide to plot versus ?
2. How will you calculate ?
3. What was your value of the lens' focal length?
4. What value for the y-intercept do you expect? Is this what your data shows?
5. What are the magnitude of error bars for the data points?
6. What is the uncertainty in your measurement of ?

1. What are some of the major errors with using Equation 2 to determine ?
2. How do the magnification values compare?
3. Do you expect the values for to be the same for each data point?
4. What is the uncertainty in your measurements of ? Does the uncertainty change during the experiment?

1. Which lens is your eyepiece and which is your objective?
2. Does it matter which lens is used as the objective and which is used for the eyepiece?
3. When the arrow is in focus, what is the distance between the lenses. Is this what you expect? Why or why not?
4. Does it appear that the telescope is actually magnifying the arrow?
5. Is the image inverted?
6. Is the image real or virtual?
7. How will you measure the apparent object and image sizes of the arrow?
8. What is the uncertainty of these measurements?
9. What is the measured value of for the telescope? How does this compare to the theoretical value?

1. What method will you use to determine the focal lengths?
2. What will you use for your object? Is your object near to you or far away? Does it matter?
3. What was your estimate of the mirror's focal length?
4. What is the uncertainty in your measurement?

1. What is your estimate of the mirror's radius of curvature?
2. What are the limitations on the placement of the object? That is, for what values of will yield real images?
3. What method will you use to determine the mirror's focal length? What, if anthing, will you plot?
4. How will you calculate ?
5. What was your value of the lens' focal length?
6. What are the magnitude of error bars for the data points?
7. What is the uncertainty in your measurement of ?
8. How will you calculate ?
9. What is the uncertainty in your measurement of ?

These Questions are also found in the lab write-up template. They must be answered by each individual of the group. This is not a team activity. Each person should attach their own copy to the lab report just prior to handing in the lab to your TA.

1. Would the lens experiment work if the lens were broken in half? Explain.

2. Under what conditions is an image from a biconvex lens inverted? Upright? Real? Virtual? Larger than the object? Smaller than the object?

3. The human eye contains a flexible lens, which focuses images onto the retina. Explain why we squint when we strain to see objects that are out of focus.

4. Does your eye form a real or virtual image? How do you know?

5. What is the purpose of the concave shape of radio telescopes and satellite TV dishes? Why are radio telescope dishes typically very large?

6. Is a "full length" mirror necessary to view your body from head to toe?

• For now we are sharing equipment with the 208 lab.
• There are a number of Objectives for this lab; you may not need to assign them all.

See the CUPOL tutorial on using the vernier caliper.