1 or 2 meter optical bench.
Illuminated object-box.
Object-image device for measuring focal length of positive lenses.
Screen, for locating real images.
Reference pointer, for locating virtual images.
Lenses, f = -20 cm.
Assorted simple lenses.
Wide focusing range telescope and holder, for locating images.
Ramsden eyepiece, or other achromatic eyelens (optional).
Short focal length achromatic lens for microscope objective.
Small specimen on microscope slide (a reticle will do.)


All students will do part A, then choose one or more of the other parts as directed.



Though the derivation of the simple magnifier formula is commonly done in textbooks, it is one of the least understood derivations in optics. For optical instruments used with the eye, especially when object or image (or both) are at infinity, is angular magnification.

The simple magnifier is a converging lens held directly in front of the eye. When we wish to examine the details of a small object, we move it closer to the eye, to make it seem larger (actually to make it subtend a larger angle in our field of vision. But when we try to bring it closer than the near point of the eye, we cannot focus on it, and it looks 'blurred'. So we instinctively move it away until it becomes clearer. It will then be at the near point of the eye, which is 25 cm for a 'standard' eye. This is shown in Fig 1, bottom. The object subtends angle α.

Fig. 1. The simple magnifier.

When we use a lens in front of the eye, we also move the object to see it `better'. We move it until its image is at the near point of the eye. Then its image will be as large as possible and still clear. This is shown in Fig. 2, top. The image subtends angle β.

The angular magnification is the ratio of the angular sizes: the size seen with optical aid, compared to the size seen without optical aid. The derivation goes like this:

          tan(β)   I/25   I   q
    m   = —————— = ———— = — = —
     25   tan(α)   O/25   O   p

Too often the important step above is omitted.

    1   1   1         q       q   -25
    — + — = —  , so   — + 1 = — = ———
    p   q   f         p       f    f

    q -25 25 q = -25 cm , so m = — = ——— - 1 , and |m| = 1 + —— p f f


Measure the focal length of the short focal length converging eyelens which you will use as a simple magnifier. Use any valid method.

Fig. 2. Measuring the magnification.

Look at the image of a reference scale seen through this lens. Move the scale until the image appears 25 cm away from the lens (use the method of parallax). Set a second reference scale at 25 cm from the lens and at such a height that it may be viewed by looking past the lens while looking at the other scale through the lens. Refer to Fig. 2, where the scale viewed through the lens is called scale A, and the reference scale viewed by looking past the lens is called scale B. If these are adjusted so there is no parallax between B and the image of A, and the heights adjusted so that the markings appear close together, the magnification may be easily obtained by directly comparing the sizes of the scale markings. In Fig. 2 the magnification is exactly 3, since the magnified view of one unit of scale A seen through the lens matches 3 units of scale B seen directly. This method of obtaining magnifications will be used in the remaining parts of the experiment, so be sure you understand it.

Derive the formula for the magnifying power of a single lens, and compare your results with this formula.


Slide and movie projectors, overhead projectors, microscope substage illuminators and enlargers for making photographic prints, all use similar optical systems. Their common features are shown below.

Fig. 3. The projection lantern optical system.

The light source is an incandescent lamp at B. Two large lenses (C) (called condenser lenses) are used to intercept a large amount of the light and redirect it in a convergent beam through the photographic transparency (O) and then through the image-forming projection lens (L). The image (I) is formed on a distant screen. The condenser lenses may be identical.

The design of a projection system proceeds as follows. The size of transparency must first be known, and generally we desire to enlarge this to fill a screen of given size. This information determines the focal length of the projection lens.

The transparency is generally an inch or more (several centimeters) in front of the condenser lenses. Since we want the full size of the transparency illuminated evenly, and the beam from the condensers should be convergent to a point on the projection lens, this will determine the necessary diameter of the condensers.

Take note of the available parts, measure their parameters, and design a projection system on paper. Calculate the performance data for the system (transparency size, screen size at a given distance, magnification, etc.) Now set up your system and check your predictions.

Problem. We wish to project an image of a 35 mm slide onto a screen 6 ft. high at a projection distance of 50 ft. The slide image is one inch high. Design completely an optical system which will do this properly. Specify all necessary dimensions and focal lengths.

Question. Consider the effect of dust particles on the (1) projection lens, (2) transparency film, or (3) the condenser lenses. In each case, what, if any, effect will the dust have on the image seen on the screen?

Puzzle. Let us define the "correct" position of the projection lens to be that position which images the picture on the film sharply on the screen. A fumble- fingered projectionist is searching for the correct position when there appears on the screen a clear image of the lamp filament (a glowing coil of wire). Does he have the lens too far from the film, too close to the film, or could it be either one? Explain, with the aid of a ray diagram.


A microscope gives very high magnification of very small objects. This requires that the object (specimen) be brightly illuminated. When the object is transparent, it may be illuminated in the same manner as in a projection lantern (see part B). In fact, in Fig. 4 we see that the imaging system of lamp, condenser lens, and microscope objective is identical to the optical system of the projection lantern. But there is one important practical difference: the microscope objective is very small and of very short focal length. It is typically only a few millimeters in diameter, and its focal length may be anywhere from a few millimeters to a few centimeters. Therefore the lamp and condenser system must be designed to produce a very small image of the lamp filament, small enough so that all the light is funneled into the small objective lens. The working microscopist may frequently change objective lenses, and this may necessitate readjusting the condenser or lamp positions for greatest light efficiency with each objective lens.

Fig. 4. The optical system of the microscope, with illuminator
The vertical scale of object and image sizes
is exaggerated with respect to the horizontal scale.

The eyelens (E) is used as a simple magnifier (see part A). So you see that the microscope is just a combination of projection lantern and magnifier. In fact, if the illumination is bright enough, and the room dark enough, the eyelens of a standard microscope may be removed, and the objective lens positioned so that it casts a real image on a screen, a wall, or the ceiling. In that case the system is optically like a projection lantern with a very high magnification.

Set up the system illustrated in Fig. 4 and investigate its properties. Measure all relevant focal lengths.


(1) Suppose a person with normal vision, using a simple magnifier, decided to adjust the position of the object so that its image was at infinity (instead of 25 cm). Derive the formula for the angular magnification of the lens under these conditions. Compare it with the standard formula (Eq. 1). Discuss.

(2) The standard microscope is designed so that the small specimen being observed is very near the objective lens, so p is only a few millimeters. The image distance of the objective lens is fixed by the position of the eyelens and by the length of the microscope tube length, standardized to 16 cm. This is the distance from the second focal point of the objective lens to the real image it forms. The instrument is focused by moving the entire tube. This design makes it is possible to assign a numerical value of magnification to each objective lens and to each eyelens. Using this information, what is the focal length of a microscope objective lens labeled 21x magnification? What is p for this objective lens?

(3) The total magnification of the microscope is calculated by multiplying the power of the eyelens by the magnification of the objective. Justify this formula and explain what meaning we can attach to this total magnification. Is it angular magnification, or linear magnification? If the magnification represents the size of the image relative to something, what is that something? Be specific.


Fig 2 is reproduced below in stereo 3D for those who are comfortable with free viewing. The center picture is for the right eye. The left pair is for parallel-eye viewing. The right pair is for cross-eye viewing.

Fig. 5. Stereo drawing of figure 2.

© 1995, 2004 by Donald E. Simanek.