THIN LENSES: BASICS
OBJECTIVE: To study and verify some of the laws of optics applicable to thin lenses by determining the focal lengths of three such lenses ( two convex, one concave) by several methods.
THEORY: Each point of a self luminous object or reflection object (O) is a source of light with a large number of rays emanating from it in all directions. A lens will alter the direction of those rays which strike it, and an image (I) of the object may be formed. If rays from a single point on the object strike different parts of the lens and eventually all intersect at some other point in space, a real image is formed. Real image formation is illustrated in Fig. 1. A virtual image is formed if the projections of the rays from a single point on the object intersect at a single point in space. A screen placed at this point will not reveal an image, but the image is visible to the brain if observed from the proper position such that the diverging rays enter the eye. See Fig 2.
A thin lens is one whose thickness is negligible in comparison to the image and object distances [object distance is measured from object to lens; image distance is measured from lens to image position]. A convex lens is thicker in the center than at the edges. Such a lens is also called a positive lens or a converging lens. A concave lens is thinner at the center than at the edges. It is also called a negative lens or a diverging lens.
The axis of a lens is the line drawn through the centers of curvature of its refracting surfaces. If a beam of rays parallel to the axis is incident on a converging lens, the beam is brought to a focus at a point on the axis called the focal point (F). If a beam of rays parallel to the axis is incident on a diverging lens, the rays diverge as though radiated from a point on the axis. This point is also called the focal point (F). The distance along the axis from the focal point to the center of the lens is known as the focal length (f). A converging lens has a positive focal length; a diverging lens has a negative focal length.
There are at least three commonly used symbols for object and image distances:
object distance: s (used here) , o (sometimes used) , p (sometimes used)
image distance: s' (used here) , i (sometimes used) , q (sometimes used)
These distances and the focal length are related by the “thin lens equation”,
1/s + 1/s' = 1/f . (1)
Object distances, image distances, and focal lengths can be positive or negative. We will see examples of both positive and negative values in this experiment.
The ratio of the image size (h’) to the object size (h) can be predicted by use of the lateral magnification factor
M_{exp}_{ }= h’ / h . (2)
[NOTE: If the image is inverted relative to the object, then h’ is negative, making M negative.]
As seen in Fig. 1, geometry gives a theoretical lateral magnification in terms of the object and image distances:
M_{the} = s' / s . (3)
[NOTE: The minus sign in Eq. (3) is necessary to keep the signs of the experimental and theoretical magnifications consistent. That is, when s’ is positive, h’ is negative.]
Part 1: Convex Lens
Three methods will be used to measure the focal length of a convex lens:

Parallel rays: If the object is sufficiently far away so that s>>f, the incoming rays will be almost parallel, and the object will be focused at approximately the focal point of the lens. This method is used in Step 1.

The thin lens equation: The image and object distances will be measured and f calculated with Eq. (1) . This method is used in Step 2. (Since object and image distances will be measured, we will also investigate lateral magnification in this step as an added bonus!)

Conjugate foci: The locations of the object and image on the axis of a lens are called the conjugate foci. For a given object distance, an image is formed at a distance determined by the thin lens equation. If the object is moved to the image location, a new image will be formed at the original object position. Thus, a lens has a symmetry with respect to its conjugate foci (s and s' can be interchanged in Eq. (1)). In the lab the position of the lens will be changed in order to effect the interchange of object and image. Referring to Fig. 3, if D = s_{ }+_{ }s' and d equals the shift in position of the lens, then one can show that
f = (D^{2} d^{2}) / 4D . (4)
This method is used in Steps 3 and 4.
PROCEDURE:

See how each of the three lenses (+5 cm convex, +10 cm convex, and –15 cm concave) acts when used as a magnifying glass. Do all the lenses magnify? Which is the "stronger" convex lens, i.e. which magnifies more when used as a magnifying glass? Which is the “weaker” convex lens?

Place only the +10 cm convex lens and the white screen on the optical bench and aim the lens at some object outside the room. (If possible, perform this part of the experiment in the hallway with the hallway lights off and with the lights in a far room on.) Move the lens or screen along the bench until the image is focused as sharply as possible. If an image can be focused, measure and record the image distance, which in this case is approximately the same as the focal length. Can you justify this assumption? [HINT: Consider the thin lens equation.]
Replace the +10 cm convex lens with the +5 cm convex lens and repeat the above procedure.
Replace the convex lens with the concave lens and repeat the above procedure.

FOR PARTS 3, 4 & 5, USE ONLY THE WEAKER CONVEX LENS.

FOCAL LENGTH. Place the slide on the optical bench. By shining a light through the slide, we will use the +10 cm convex lens to form an image of the slide on the screen. Hence the slide will become our object in this part. Next place the lens at some position beyond the focal length from the object and focus an image of the object on the screen by moving the screen. Measure object and image distances, s & s', and calculate the focal length, f.

MAGNIFCATION. The length of one of the object arrows is h = 30 mm. Measure the length of the image arrow, h’. (If you cannot image the entire arrow clearly, then use the large or small circle as your image and measure the diameter of this image circle. This diameter is the value of h’. The object circles have diameters of 20 mm and 10 mm. Use the appropriate value for h.) Note whether the image is inverted and use the proper sign for h’. Calculate the experimental and theoretical magnifications using Eqs. (2)&(3) and compare the values.
Repeat these measurements and calculations for at least two more object distances. Be sure to have at least one case where the image is magnified (bigger than object) and at least one case where the image is demagnified (smaller than object). Average your focal length values.

Referring to Fig. 3, if the image and object distances are equal (s = s’), then D = 2s, d = 0 and Eq. (4) gives f = s/2. (The same result is found from the thin lens equation.) This would put the image a distance of 4f from the object. See if this is true experimentally. Using the average value of f from Step 2, place the lens a distance of 2f from the object and move the screen until the image is formed. Measure the distance between object and image. Is it 4f? Record the difference between 4f and the measured object to image distance.

Place the screen at a distance from the object that is greater than 4f. Call this distance D. By moving the lens, find the two lens positions that give sharp images. Measure the distance between these two lens positions. This is distance d. Using Eq. (4), compute the focal length and compare with the values obtained above previously.
REPORT:

For each of the procedures above: draw a diagram, record your data, show your calculations, state your results, and answer any questions posed. In each case, be sure to include a discussion of possible errors and the estimated accuracy of your data and how this uncertainty in data affects your results.

Compare the values of the focal length from the four procedures. Comment on whether they agree with one another within experimental uncertainty.

Comment on your values for magnification. Is the theoretical value the same as the experimental value within experimental error? Is there just one value for magnification for this lens? If not, what does the magnification depend upon?
Part 2: Concave Lens
The method of determining the focal length of a concave (negative, diverging) lens will involve using it in conjunction with an auxiliary convex (positive, converging) lens. In Fig. 4, I_{1} is the image of the object O_{1} which would be formed by lens #1 if the diverging lens #2 where not present. But with lens #2 in place, the rays do not converge as quickly, and the image I_{2} is formed farther away. When analyzing multiple lens systems, the image formed by the first lens is considered to be the object of the second lens (I_{1} O_{2}), and the thin lens equation is used.
PROCEDURE:

IMAGE AS OBJECT.

Before we consider a concave lens with a convex lens, let's work with two convex lenses to see more clearly how to work with two lenses. Start with the weaker of the two convex lenses and focus an image on the screen. Try to get an image that is somewhat close in size to the object. Record the object distance, image distance, object height, and image height. Calculate the focal length of this weaker lens. Compare to the values you have already obtained for it in Part 1.

Next place the stronger convex lens about 10 cm beyond the image of the first lens and record this position. The distance from the screen to the second lens will be the object distance for the second lens. Now relocate the screen so that it is beyond the second lens and see if you can focus an image. If you can, record the position of the screen and the final image height, and calculate the image distance for this second lens. Now calculate the focal length of this second (stronger) lens. Should the stronger lens have a focal length bigger or smaller than the thinner lens? Does it? Is this focal length close to what you determined in Part 1, procedure 1 (hallway method) for this stronger lens?

FOCAL LENGTH OF CONCAVE LENS.

Remove the stronger convex lens and form an image with the weaker convex lens, just as you did above. Record the position of the image since this will become the object for the concave lens. Without moving either the object or convex lens, place the concave lens on the bench BETWEEN the convex lens and the screen. (NOTE: We cannot position the lens behind the image since a concave lens does not converge the light diverging from the image the way a convex lens does. Instead, we will use the concave lens to diverge the already converging light coming from the convex lens before it forms an image. If the concave lens is weaker than the convex lens, the rays will still form an image but further away as shown in Fig. 4.)

Since the object for the concave lens (image from the convex lens) is on the "wrong" side of the lens, we must make the object distance for the concave lens negative. Now reposition the screen until a new image is formed. Record the position of the concave lens and of the new image. By remembering that the initial position of the screen is the location of the virtual object of the concave lens, compute the focal length of the concave lens. A concave lens should have a negative focal length. Does yours?

Repeat the above using a different distance between the object and the convex lens. Compare your two values for the focal length of the concave lens. Are they the same? Should they be?

MAGNIFICATION.

Using h'_{1} (image size of convex lens) as the object size for the concave lens (h_{2}), determine the magnification (both experimentally and theoretically) for the concave lens. Are the object and image both upside down? Should M_{concave} be positive or negative?

Now determine the experimental magnification of the system of lenses. This is the final image size (h'_{2}) divided by original object size (h_{1}). Does M_{final} = M_{convex}_{ }+_{ }M_{concave} or does M_{final} = M_{convex }*_{ }M_{concave} ?
REPORT:

For each of the steps above: record your data, show your calculations, state your results, and answer any questions posed. In each case, be sure to include a discussion of possible errors and the estimated accuracy of your data and how this uncertainty in data affects your results.

Show from theory that M_{final} = M_{convex }*_{ }M_{concave}. (In fact, for a system with N lenses, the final system magnification is simply the product of all of the individual magnifications, .
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