## L-2 THIN LENSES
To investigate the relation between focal length, object distance, and image distance for converging and diverging thin lenses; and to determine their focal lengths. The experiment also provides experience in using parallax methods to locate images in space.
Optical bench, three lens holders, four carriages, illuminated object box, screen, positive and negative lenses, rod or pencil for use as a reference pointer, object-image screen and holder, mirror.
(1) (a) For convenience of discussion we assume that the light passes through the lens from left to right. Ray diagrams will follow this convention. (b) The
(c) The
(d) A lens which (e) A lens which (f) Light rays from a point source (object) passing through a lens emerge convergent to
a point or divergent from a point. In either case, that point is called the (g) When the emergent rays (h) When the emergent rays (i) The distance from the object to the lens is designated p. (j) The distance from the image to the lens is designated q. (k) When rays (l) When the rays (m) When the object is to the left of the lens, and the image to the right, both p and q are positive. If either one is on the other side of the lens, the distance of that one is negative. (n) Finally, using these conventions, all thin lenses obey the lens equation:
The graph of the thin lens equation is an hyperbola. Its asymptotes cross the p and q axes at values equal to the focal length. Fig. 6 shows the object-image relation for a converging (positive) lens. The hyperbola has two branches, symmetric about the point (F,F).
(2) real image.If rays emerging from a lens are divergent, one may see the image by looking back
through the lens, toward the source. The image appears as a luminous point, floating in space,
but located at a different place than the actual object. Such an image is called a A luminous surface may be thought of as an infinite array of luminous points. Lenses can
form images of such surfaces. Whether the source is a point, or a surface, it is called the
(3) parallax.The brain processes these two images, translating the parallax differences into a sensation
of depth, solidity, and relative location of objects in space. This process is called stereoscopic
vision. While stereoscopic parallax gives very good We also use parallax in another way. When we move about, our eyes see our surroundings from a continually changing point of view. This provides the brain with additional depth information, and is particularly striking when we observe a scene from a moving vehicle. Even a one-eyed person can, by moving the head, observe objects from different angles to make depth and distance judgments. A demonstration of parallax will be set up in lab, consisting of several objects of unknown size at different distances. With no size or shadow clues, you cannot tell their relative distances using only one eye keeping your head stationary. But when you are allowed to move your head back and forth you see the objects shift relative to each other. If, as you move your head to the right, object A moves to the right relative to object B, then object A is farther way from you than object B.
Objects at the same distance from the eye show no relative shift; they maintain a constant lateral separation as your head moves. Our eyes are quite good at detecting the presence or absence of relative motion, so we can use parallax as a sensitive means for locating two objects at the same distance, even when one object is a "phantom" like a virtual image. To do this, we arrange a reference object on a scale, and move it until there is no parallax shift between it and the image. We say we have then "eliminated parallax" and our reference object is at the same distance as the image. It may also be used for real images, but there are simpler and more accurate methods available for locating them. When this method is used for locating images seen through lenses, the lateral motion of the eye must be kept very small, so the eye is always looking through the center portion of the lens. If the eye sweeps across the lens from edge to edge, the image appears to warp in shape due to the distortions and aberrations of the lens. This may easily be confused with the parallax shift.
An optical bench is a device for testing optical components and systems. It allows positioning optical elements, moving them along a straight line, and measuring their positions accurately. An object box, usually with internal illumination, provides a luminous pattern on a plane surface. Lens holders position and center lenses. Screens are flat surfaces which may be opaque (either painted black or white), or they may be frosted glass. Screens are used to locate real images. All of these components are attached to sliding carriages which move them along the bench and provide a reference pointer which indicates the carriage position along a metric scale. It is important that all lenses be aligned along a common axis parallel to the bench scale. For most purposes, the plane of the object and the plane of the screen should be parallel to each other and perpendicular to the lens axis. This alignment and centering should be attended to first, for the ultimate accuracy of your results depends upon it.
There are Move the screen about 5 cm toward the object and repeat. Continue bringing the
screen closer in increments, locating two images for each screen position. Eventually
the screen is so close to the object that only one image is found, and if the screen is
moved any closer, no
Observation of virtual images is much easier if you move the lens to the rightmost end of the bench, and have that end near the edge of the table so you can comfortably place your eye near the lens. A pencil or rod mounted upright in a bench carriage can serve as a reference-pointer to aid in locating the image by the method of parallax. Move the object closer to the lens, less than one focal distance. Accurately locate a few images in order to test whether they fit the lens equations. If the image is beyond the end of the bench, obviously you can't get an accurate location, so move the lens forward or back until you get a case where the image falls within the limits of the bench. Place the reference-pointer about where you think the image is. Place your eye so you
can see the image through the lens, and also see the pointer directly. Ignore the image of the
pointer seen through the lens. Fig. 8 shows approximately what you should see. Now move
your head back and forth This process is easier to do than to describe. If in doubt whether you are doing it correctly, ask your instructor to demonstrate it and check your results.
Note that it is (3) (4) To create such a situation we may use a converging (positive) lens to re-converge the rays coming from an object box. The lens being studied is then placed in this convergent beam. Set this up as shown in Fig. 10.
Lens B is the one being studied; use the The first lens (A) creates a real image in space. Measure its position when lens B is
removed from the system. You needn't measure the positions of lens A and the object box,
you only need the position of the (5) In similar fashion, use the negative lens of part (3) with a virtual object. Use this arrangement to study two cases: (i) real final image, and (ii) virtual final image. See whether the results agree with the lens equation and with your previously determined focal length for this lens. The real image data is probably the best data for this lens, from which you can get the best determination of its focal length. (6) The telescope is rigidly mounted on a sliding carriage and aimed parallel to the bench
to receive the light emerging from the lens system. With left-to-right passage of light,
the telescope must be to the right of the image. Focus the telescope on the image, then
adjust its cross hairs or reticle to eliminate parallax. Now,
(7) (8) a) Use the object-image screen (see part 6). This works only for positive lenses. b) The focal length of a positive lens may be found by aiming it at a distant scene, allowing the image of that scene to fall on a screen. The lens-screen distance is then approximately the focal length. c) The focal length of a negative lens may be checked by aiming it at the sun.
d) Theory predicts that the diopter rating of two
is a straight line! Furthermore its intercepts
are at 1/p = 1/q = 1/f, providing a neat way to use all of your best data for a particular
lens to determine the focal length. Do this two ways:
(1) Directly plot the reciprocals of the data points and use a ruler to obtain the best straight line fit. Average the intercepts and take the reciprocal to obtain f. (Fig. 13) (2) Do a least squares fit (linear regression) of the reciprocals of the data points. A computer program is available which makes this job easy. It gives the intercept and slope of the best straight line fit, from which you may obtain the other intercept, then average these to get the reciprocal of f. Do the above for each lens. Use all of the data, for both real and virtual images.
(1) How does the graph for a diverging (negative) lens differ from that of the converging lens shown in Fig. 6? Sketch the graph, accurately showing shape and its intercepts. Do the same for the reciprocal graph, like Fig. 13. (2) Explain why only one real image is found at the critical object-screen distance, and none at smaller distances. What is the relation between this distance and the lens focal length? Use the lens equation to prove this relation algebraically. (3*) Explain why the method described in section 8 (c) works for finding the focal length
of a negative lens. Use geometry (draw a correct and complete ray diagram) to derive the
relation which validates this method. What is special about the choice of a circle
(4) What optical device has a focal length of infinity? Describe it, and explain the reasoning which led you to discover it. Likewise, describe an optical device which has a focal length of negative infinity. (5*) Make sketches like Figs. 6 and 12 for both positive and negative lenses (four
sketches total). Label the segments of each graph by kinds of objects and images (real or
virtual) obtained. Note the correspondence of points between the hyperbolic and straight line
graphs. Points at infinity are mapped into finite points and vice versa by the reciprocal
transformation. Sometimes mathematicians say that +infinity and -infinity are not two points,
but the same point" (6) Prove that a diverging lens always gives a virtual image of a real object and can only produce a real image when the object is virtual and inside the focal point. (7) Prove that a converging lens always gives a real image of a virtual object, and can only produce a virtual image when the object is real and inside the focal point.
The 3d drawing above shows light rays from object to image. The object (O) is seen in the picture at the left rear. The lens (L) projects an image (I) onto a screen at the near right. This stereoscopic drawing has the image for the right eye in the middle. Those who use parallel (wall-eyed) viewing must use the left two drawings. Those who can do cross-eyed free viewing may use the right two drawings, or, they may use the enlarged inverted stereo pair shown below. Those who need instruction in 3d viewing may consult my 3d illusions document. The drawing is in perspective, so even if you can't view stereo, the arrangement of components should be apparent. Here's a larger version just for cross-eyed viewers. Figures 8 and 9 are reproduced below in stereo 3D for those who are comfortable with free-viewing. Text and diagrams © 1994, 2004 by Donald E. Simanek |