Lenses and Telescopes

A. Using single lenses to form images

The simplest variety of telescope uses a single lens. The image is formed at the “focus” of the telescope, which is simply the focal plane of the lens. A piece of film or – more commonly these days – an electronic detector known as a “charged-coupled device,” or “CCD” is placed in the focal plane, and a picture is taken. In this exercise we’ll find out about the properties of a lens that affect the size and brightness of the image it forms.

Each group will have a set of 6 lenses for this exercise. We also have a few large-diameter lenses at the front of the room that we will need to share among the groups. The lenses have different diameters and they are curved by different amounts. The two key properties of a lens are its diameter and its curvature.

The diameter is simply that: the diameter of the lens, viewed face-on. You can measure it with a ruler. The diameter is important because it determines how much light a lens collects. The larger the diameter, the more light it can collect and focus onto a screen or your eye.

The curvature of a lens determines how strongly it bends incoming rays of light. Rays that pass through the center of the lens aren’t bent at all. Rays that pass through the edges are bent inward. The more curved the lens, the more these rays are bent. The lenses we are using are called “spherical” lenses. That means that the surface of the lens is a part of a sphere (see diagram below). The more curved the lens, the smaller the sphere it would fit into. Note that two lenses of the same “curvature” but different diameters would be of different thicknesses. The larger diameter lens would have a greater thickness than the smaller diameter lens, even if it had the same curvature. Keep these ideas in mind as you are doing the exercises below.

larger circle high curvature lens small circle lower curvature lenses, both with the same curvature since they fit inside the same circle

NOTE: Please handle the lenses carefully--don’t drop them, and try to hold them by the edges so as to keep them free of fingerprints. Step 1: Use a ruler to measure the diameter and focal length of each of the lenses at your station. (Each lab partner should take turns measuring the properties of the lenses.) To determine the focal length of a lens, use the lens to project an image of a distance object on a piece of paper. Move the lens toward and away from the paper until you find the position that produces the sharpest image of the distant object. Then, have another group member use a ruler to measure the separation of the lens from the paper. Measure from the paper to the edge of the lens. Record your results in Table 1 on the worksheet, matching the lenses to the letters listed in column 1.

Q1: Judging from the results in Step 1, what property of a lens would you say determines its focal length: its diameter or its curvature? Check your answer by testing a pair of lenses with very different focal lengths, then by testing a pair with the same focal length.

Step 2: You may have noticed that different lenses form images of different sizes on the screen. With the help of a partner, use a ruler to measure the width of the projected image for each of the lenses and record your results in Table 1.

Q2: Judging from your results from Step 2, and using the lens data in Table 1, which property of a lens determines the size of the image that will be formed by the lens: its diameter or its curvature? Check your answer by testing a pair of lenses with similar image widths, then by testing a pair with very different image widths.

Step 3: Lenses are often characterized by a third quantity (in addition to diameter and focal length) known as “focal ratio.” The focal ratio of a lens is the ratio of the lens’ focal length to its diameter. To determine the focal ratio of a lens, you divide its focal length by its diameter. The result is a pure number (i.e. with no units). Compute the focal ratio of each of the lenses in your set and enter the results in Table 1 on the worksheet.

Now list the lenses by letter, in order of their focal ratio (largest to smallest).

Q3: List of lenses, ordered by focal ratio

Now use each lens, in turn, to make an image on the screen. Start with the lens with the largest focal ratio, and proceed until you’re making an image with the lens with the smallest focal ratio. As you make the images, pay attention to how the lens concentrates light into a small or large image.

Q4: Which lenses make more concentrated images, those with the largest focal ratios, or those with the smallest focal ratios?

Q5: Is there any way that a lens with a smaller diameter can make a more concentrated image than a lens with a larger diameter? Explain your reasoning.

Step 4: One of the reasons we use telescopes is because they can gather more light than our eyes. A lens that can gather more light will create a brighter image, which is easier to see with our eyes. We talked about concentration in the last step, but we should also consider the total amount of light being gathered by a lens.

Q6: Will a more concentrated (or more intense) image have a smaller image size or a larger image size?

Q7: Which lens has the most concentrated (most intense) image?

Q8: Which lens has the least concentrated (least intense) image?

To see what feature of a lens determines the overall brightness of the image that it will form (regardless of image size), pick two or three lenses that have about the same image size.

Using only these lenses, compare the images formed by each. Determine which lens forms the brightest image, and which forms the faintest one.

Q9: List the lenses in order of image brightness from brightest image to faintest image.

Q10: Which property of the lenses seems to determine how bright the image that is formed will be? Justify your answer by referring to the results of your experiment with the lenses.

Q11: What do you think can account for the results regarding image brightness above? Explain as precisely as you can exactly what you think is going on to make one lens make a brighter image than another.

B. Making and testing a simple 2-lens telescope

One of the simplest telescopes to make is one with two lenses of the type studied in Exercise 1 (known as “converging” lenses because they cause parallel rays to converge or come together to a point). The first telescopes ever used to study the heavens – those built by Galileo in 1610 – were 2-lens telescopes. Telescopes that use lenses are called “refractors” (refracting means to bend, these are telescopes that bend light rays). Many modern telescopes use mirrors instead of lenses to bend light, in part because they are easier to manufacture in large sizes. Telescopes that use mirrors are called “reflectors” because they reflect light. In this exercise, you will build a refractor and test its properties. Your lens kit also contains a mirror, which you can use to form a simple reflector telescope as well. The principles of operation of reflectors are very similar to those of refractors.

In a refracting telescope, the first lens (the one the starlight hits first) is called the objective lens. The second lens (the one in front of your eyeball) is called the eyepiece lens. An easy way to remember this is that the objective lens is nearest to the object, which the eyepiece lens is nearest to your eye. The objective lens in a telescope is always much bigger than the eyepiece lens (why do we want a large-diameter objective lens? Think about Q10 and Q11 from Exercise 1). In a reflector telescope, a mirror (also called the “objective”) replaces the objective lens. Refractors and reflectors both use lenses for eyepieces.

The objective lens or mirror in a telescope is a permanent (non-removable) feature of the telescope. The most important thing about the objective is its diameter. The larger the diameter, the more light the telescope can collect and focus, and the brighter the images it produces. The light-collecting capacity of a telescope also enables faint stars to be seen through a telescope that cannot be seen with the unaided eye. The other important feature of the objective is its focal length. The longer the focal length, the larger the image formed in the focal plane.

The eyepiece lens in a telescope acts like a magnifying glass, magnifying the image in the focal plane so that the observer can see its detailed features. Eyepieces are typically interchangeable on a telescope. They range in size from the size of a 35mm film canister to the size of a telephoto lens on a camera.

Interestingly, the smaller the focal length of the eyepiece (the more curved it is), the more magnification it provides. The telescope magnification formula is:

magnification = (focal length of objective) / (focal length of eyepiece) where the focal lengths of both lenses are expressed in the same units (typically mm).

Beware however that this doesn’t always mean a small focal length eyepiece is better! The larger the magnification, the smaller the patch of sky (“field of view”) you will get to see through the telescope. To get a large “field of view” (to see a bigger patch of the sky) you need to use low magnification. Most telescopes enable you to see at most a patch of the sky about 1 degree on each side! Moreover, because the Earth’s atmosphere is turbulent, which blurs out the images of stars, there are limits to how much you can magnify a star image before it just looks like a big fuzzy blob.

Now let’s build a 2-lens refracting telescope and see all this for ourselves.

For this exercise, we’ll use four lenses: two with larger diameters (we have limited numbers of these, so they are not in your lens kits. You will need to come borrow them individually from the front of the room) and two with smaller diameters (see the board for which lenses to use).

Q12: Based on their relative curvature, which of the two large-diameter lenses do you expect to have the longer focal length?

Q13: Based on their relative curvature, which of the two small-diameter lenses do you expect to have the longer focal length?

Check your answers to Q12 and Q13 by projecting images with both sets of lenses.

Now make a simple telescope with two lenses by holding the lenses in your hands. Use one of the larger-diameter lenses as the “objective” lens. Hold it at arm’s length, roughly, and look through it at something in the distance (it won’t look magnified yet). Then use one of the smaller-diameter lenses as the “eyepiece” lens. Place it close to your eye, in front of the objective lens. If the distant object isn’t in focus, experiment with changing the distance at which you hold both the objective and eyepiece lenses (one at a time).

Ask your instructor for assistance if you’re having trouble with your telescope!

Once you’ve gotten the hang of one hand-held telescope, try different objective lenses (either large-diameter lens) and different eyepiece lenses (either small-diameter lens). There are four different possible combinations of lenses from which to make telescopes. Try them all!

Q14: Consider the four possible combinations of objective and eyepiece lenses. Given the formula for magnification on the first page of this exercise, compute the magnification each combination should produce. Record the relevant focal lengths and magnifications for each lens combination in Table 2 on the worksheet.

Q15: Now try building the two telescopes that should provide the most and least magnification. Pick something on the other side of the room to look at. Can you visually confirm that one provides more magnification than the other?

Q16: Which of the two telescopes in Q15 provided the larger field of view? The one that gave the most magnification, or the one that gave the least magnification?

Lenses and Telescopes Worksheet

Table 1: Lens Properties Lens Lens Sketch Diameter Focal Image Focal (edge view, roughly to scale) (cm) Length Size Ratio (cm) (cm) A

B

C

D

E

F

Table 2: Telescope Properties

Objective Focal Length Eyepiece Focal Length Magnification: lens Objective lens Eyepiece (cm) (cm)