Lecture 26Lecture 26Ch. 34Ch. 34Physics 2102Jonathan DowlingOptics: Images Optics: Images —— LensesLensesThin LensesThin Lensesfip111=+For small angles and thin lenses, Convergent: f positiveDivergent: f negative!"#$%&''=2111)1(1rrnfLens maker’s equationConvergent lensDivergent lensLocating Images by Drawing Rays1. A ray of direction initially parallel tothe axis will pass through the focalpoint.2. A ray that initially has a directionthat passes through the focal point willemerge parallel to the central axis.3. A ray going through the center of thelens will be undeflected. The image of a point appears whereall rays emanating from a pointintersect.• An object placed beyond a convergent lenses’ focal point, will producea real, inverted image on the other side of the lens. (This is theprinciple used in movie projectors).• An object placed between a convergent lens and its focal point willproduce a virtual image on the same side as the object. (Contact lensesfor farsightedness, magnifying glasses.)•Divergent lenses always produce a virtual image on the same side asthe object (Contact lenses for nearsightedness).• Real images have image distance + i, virtual images have –i.ExampleExample• An object 1.2cm high is placed 4cm from a bi-convex lens withr1=10cm and r2=15cm. Find the position and size of the image.• A second lens of focal length +6cm is placed 12cm to the right ofthe first lens. Find the position and size of the new image.ExampleExample• An object 2cm high is placed 4cm from a bi-convex lens withr1=10cm and r2=15cm, and index of refraction n=1.5. Find theposition and size of the image.• A second lens of focal length +6cm is placed 12cm to the right ofthe first lens. Find the position and size of the new image.The human eyeThe human eye consists of avariable-geometry lens (crystalline)which produces a real image on a“screen” (retina) which is transmittedto the brain via the optical nerve.The cristalline automatically adjusts itself so we see well any objectplaced between infinity and a distance called “near point” (about 25cmfor a typical 20 year old). The “image distance” is the eye diameter~2cm.Optical Instruments: the Human EyeOptical Instruments: the Human EyeCombination of Several (Thin)Combination of Several (Thin)LensesLenses: The MicroscopeF1F1F2F2If lenses are very close, the compound lens has 1/f~1/f1+1/f2Eyeglasses & Contact LensesEyeglasses & Contact LensesA farsighted person needs a convergent lens.A nearsighted person needs a divergent lens.The “power” of a lens is measured in dioptres: P=1/f with f is in m.Glasses with -6D are divergent glasses with f=−1/6D =−0.17m=−17cmThe dioptres add! Two lenses have 1/f=1/f1+1/f2 → D=D1+D2The magnification of an object is m=i/p=iθ/h,but i=eye diameter.Maximum magnification: m~2cm/25cm (!?)Angular magnification (different from lateral): mθ=θ’/θ. fcmfhcmh 25m ' 25=!="""Magnifying LensMagnifying LensWe’d like to make p smaller (move the object closer). We use amagnifying lens to produce a (larger) image than our eye can see:Very near the focus!Microscope:To increase the magnification of a lens, one wants to have a shortfocal length. That means small radii of curvature (very curved lens).This, in turn implies a lot of aberration (one is immediately out ofthe thin lens approximation). A solution to this is obtained bycombining two lenses. The resulting device is called microscope.pim !=Object O is magnifiedby the objective:And its image is magnifiedby the eyepiece:fcmm25=!Total magnification:eyobfcmfsmmM25!=="Refracting Telescope:Telescopes are arrangement of lenses that improve vision of objectsvery far away. They are configured like a microscope. However, theobjective forms an image essentially at its focus, and therefore theeyepiece’s focus has to be placed at that same point.The magnification is given bythe ratio θey/θob, and sinceobobfh /'=!eyeyfh /'=!eyobffm !=Refracting telescopes are of limiteduse (chromatic aberration). Reflectingtelescopes built with mirrors are preferred in astronomy.ExampleExampleThe world’s largest refracting lens telescope is at the YerkesObservatory of the University of Chicago at Williams Bay,Wisconsin. (Bigger telescopes use mirrors instead of lenses.) Theobjective has a diameter of 102cm and a focal length of 19.5m.The focal length of the eyepiece is 10cm. What is its magnifyingpower?1951.05.19!=!=!=mmffmeyobWhy so large (102cm)? Because thelarger the objective, the more light itgathers.Reflective TelescopesReflective TelescopesKeck observatory (MaunaKea, Hawaii) and theHale-Bopp comet.Largest optical telescope,composed of 36 (!)hexagonal mirrorsegments performing as asingle mirror 10m
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