152-MPX Manual - Teiescopes and Combinations of Lenses I...

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Unformatted text preview: Teiescopes and Combinations of Lenses I. OBJECTIVES i. Measure the magnification of an astronomical telescope. ii. Measure the magnification of a terrestrial telescope. iii. Measure the effective focal length of a combination of two thin convex lenses in contact. iv. Study the image magnification of a combination of lenses. 1]. EQUIPMENT Incandescent light source, optical bench, standard component carriers, from the PASCO optical component collection: Component carriers (4), lenses: f = 127 m, f = 18 m, f = -22 m, f = 252 m, f = 48 mm, screen, arrow target, auxiliary lamp, telescope target. A photograph of the equipment is shown in the appendix of the Geometrical Optics experiment. III. INTRODUCTION In this section we discuss briefly the optical set-ups which you will study. 1. Astronomical telescope A schematic diagram of the astronomical telescope is shown in Fig. 1. (—— d ll '1'. '1 I ”ll VIII/III. L2 Eye (Ocular) Fig. 1. Schematic diagram of the astronomical telescope. 61 62 MODERN PHYSICS LABORATORY MANUAL FOR PHYSICS 207 It consists of two convex lenses L1 and L2 of focal lengths f; and f2, respectively, placed at a distance d = f; + {2. Lens 1.; is known as the objective and L2 as the ocular. The object is assumed to be at a very large distance and practically is considered to be at infinity, a very good approximation for stars. Lens L1 forms a real and inverted image of the object at its focal point B. The magnitude of this intermediate image is equal to h' = BC. The intermediate image becomes the object for lens L2. Since the object lies at the focal point of L2, the image of it will be virtual, with the same orientation, and appear at infinity. Thus the final image is inverted with respect to the original object. The absolute value lml of the magnification is given by mg, m where 91 and 92 are shown in Fig. 1. If we assume rays propagating close to the telescope axis we have that: 01 << 1 and 92 << 1. From triangle ABC we get: 01 :- tan 0; = h'Ifl. (2) From triangle BCD we have: ozatan92=h'/f2. (3) If we combine equations 1, 2, and 3 we get: ##le- (4) The sign of m is negative. This indicates that the final image is inverted. 2. Terrestrial telescope The terrestrial telescope consists of a convex objective 1.; of focal length f1 and a concave ocular L; of focal length {2. They are placed at a distance d = fl - If; I. A schematic diagram is shown in Fig. 2. L 1 (Objective) Fig. 2. Schematic diagram of a terrestrial telescope. TELESCOPES AND COMBINATIONS OP LENSES 63 The objective lens L1 forms an intermediate image of height h' = BC at the focal point B of L, which now lies to the right of L2. This image acts as a virtual object for [4. The final image formed by L2 is virtual, upright and at infinity. The magnification m is given by eq. 4. It has a positive sign which indicates that the final image is upright. 3. Combinations of Lenses (a) Fig. 3. Consider the combination of lenses shown in Fig. 3. Two convex lenses in contact. It consists of two thin convex lenses 1.; and 14, which have focal lengths f, and f2 respectively. The two lenses are in contact at point C. In Fig. 3 A is a point light source, and B its image formed by the two lens combination. Lens L1 forms an intermediate image of A at a distance i]. to the right of Ll. This intermediate image becomes the object for lens L2. The position of B can be found by applying Gauss's law forlensesforLlanqu. L1: .J_+.L=L, (5) AC i1 f1 . The intermediate image, located at 02 , will act like a virtual object for L; with 02 = 'i‘lr L2: i+_1_=.1.l 02 BC f2 (6) or __L+L=_1_, i1 BC {2 (6a) IfweaddequationsSandeeget 4.4-; 21-4-1.- (7) AC BC f1 f2 Equation 7 is Gauss's law for a lens with effective focal length f given by }=§-+é- (8) 1 This shows that the combination of L1 and L2 acts like a single lens with effective focal length f given by Eq. 8. This effective focal length for two converging lenses is always smaller than the focal length of either of the two lenses. 64 MODERN PHYSICS LABORATORY MANUAL FOR PHYSICS 207 (b) Consider the lens combination shown in Fig. 4. <—> f1 <---)(———-—--> d <—————»'4—-> f2 4—) Fig. 4. Two convex lenses at a separation d. The object is a light point source A placed now at the focal point of L1. Light will emerge from L1 as a parallel beam. Lens L2 is placed at a distance d from L1 to form the final image. Distance d can vary from 0 to any length. The magnification of this I'M-if- (9) This lens combination is useful in optical experiments because it forms a parallel light beam of variable length d. For example optical components (for example polarizers, attenuators, etc.) can be inserted in Section CD without any modification of the set up. References Any introductory physics text. Key words: TelescOpes, lens combinations. IV. EXPERIMENTAL METHOD The various optical components used in this experiment have been described in the Geometrical Optics section. Below we describe the procedure used to measure the magnification of the astronomical and terrestrial telescopes. The viewing target used for this purpose is shown in Fig. 53. TELESCOPES AND COMBINATIONS OF LENSES 65 Target .V I \V I “‘\‘ l 1“ V L\‘\‘ L —_ L R __ L —__ L * — L —— L L —_ R L Fig. 5(a): Lined target for telescope. Fig. 5(c): Image (darker lines) is seen as superimposed on target lines. It consists of a series of parallel equidistant lines, and we will refer to it as the target. It is placed at a distance D from lens L1, along the telescope axis so that the condition D>>d is satisfied so that the target can be considered to be at infinity (see Fig. 5b). In Fig. 5b a top-view of the set-up is shown. The observer's right eye is used to view at the target through the telescope. The observer can look directly at the target using his left eye. If the left eye is closed, then only the magnified image of the target is observed. The magnified lines are shown as the heavy lines in Fig. 5c and are labeled by the letter R. If both eyes are kept open, the two target images are superimposed on the observer's field of view as shown in Fig. 5c. The target lines viewed through the unaided left eye are labeled by the letter L. The magnification l ml is given by the spacing of the L-lines divided into the spacing between the R-lines. For example in Fig. 5(c) l ml = 5. Note, however, that this ratio may not be an integer in general. fi MODERN PHYSICS LABORATORY MANUAL FOR PHYSICS 207 V. PROCEDURE 1. Astronomical telescope. Use the set-up shown in Fig. 6. Target Optical Bench Fig. 6. Arrangement of lenses, Optical bench and target for the telescope observation. Use an objective lens 1.; with foal length fl = 127 nun and an ocular lens L2 with f2 = 18 mm. Place the carriage component of 14 at one end of the optical bench and set the distance d between L1 and L2 equal to f, + £2 = 145 mm. Position the target at a distance D>>d along the telescope axis, i.e.D-=300cm, d=10cm. Adjust lens L; by moving its carrier along the optical bench back and forth until a sharp image of the target is observed. The angle of the image can be adjusted by slightly tilting L; on its carrier. Follow the procedure of Section IV to determine the telescope's magnification. Record its value in your notebook. 2. Terrestrial telescope. Repeat the procedure of Section V1 for the terrestrial telescope. Use: f1: 127 mm and f2 = - 22 mm (a comve lens). Set (1 = fl - lle = 105 mm. Determine the telescope's magnification. Record its value in your notebook. 3. Use two lenses L, and [q with focal lengths f1 = 127 mm and f2 = 252 mm in the set up of Fig. 7. 3:“ L1 1.2 Optical bench El lamp |(——o—>j(———i——>| Fig. 7. Set up for a combination of two converging lenses in contact. See also diagram in Fig. 3. The two lenses are mounted on a common component carrier. Form the image of the incandescent lamp filament on a screen. Measure the object-lens center distance (0) and the image—lens center distance (i). Take into account that the light bulb filament is recessed from the lamp's front end by 30 mm. Vary the distance 0 between 110 and 180 mm in steps of 5 mm. Form a sharp image on the screen and determine the corresponding image distance i. Record both 0 and i in your notebook. TELESCOPES AND COMBINATIONS OF LENSES 67 Use the set-up of Fig. 8 with f1 = 48 mm and f2 = 252 mm. Arrow target Optical <—t,——><-—d——><———r2———> Fig. 8 Set up for a combination of two converging lenses at a separation d. See also diagram in Fig. 4. The distance d is variable. Use d = 300 mm. Place L1 at a distance f1 from the arrow target and position the screen at a distance f2 from L2. Adjust slightly the position of L1 along the optical bench to get a sharp image of the arrow target on the screen. Determine the magnification lml by counting how many screen divisions fit between two successive divisions of the arrow image. Record | ml in your notebook. VI. FOR THE REPORT 1. Calculate the magnification lml of the astronomical telescope using Eq. 4. Compare it with the experimental value of Section VI. Repeat the previous step for the terrestrial telescope studied in Section V2. Tabulate the data of Section V3. Add columns for 1/0 and l/ i. Plot [/0 versus 1/ i. From eqs. 7 and eqs. 8 the points should fall on a straight line of slope = -l and intercept = l/ f. Use the least squares method to calculate the intercept and from its value, determine the lens combination effective focal length f. Use eqs. 8 to calculate f. Compare the calculated and experimental values. Use eq. 9 to calculate the magnification lml for the experiment of section V4. Compare the calculated and experimental values. VII. QUESTIONS 1. The images of the astronomical and terrestrial telescopes used in this experiment are colored around the edges. Explain that. How can this "chromatic aberration" be avoided in practice? An optical lab stocks a large collection of lenses with both positive and negative focal lengths. A lens with positive focal length larger than the largest available positive f lens is urgently needed. Suggest how this can be achieved using a combination of available lenses. Show that for the arrangement of lenses in Fig. 4, the magnification is given by =Iz, ln'l f] ...
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