1. Example: What kind of lens must be used, in order to give an erect image 1/5 as large as an object placed 15 cm in front of it? M = -q/p  -q/p=1/5 So q = -p/5 = -15/5 = -3 cm 1/p + 1/q = 1/f  1/15 - 1/3 = 1/f 1/f = (1-5)/15 f = -15/4 = -3.75 cm Diverging lens

2. Magnifier Consider small object held in front of eye Consider small object held in front of eye Height yHeight y Makes an angle  at given distance from the eyeMakes an angle  at given distance from the eye Goal is to make object “appear bigger”: ' >  Goal is to make object “appear bigger”: ' >  y 

3. Magnifier Single converging lens Single converging lens Simple analysis: put eye right behind lensSimple analysis: put eye right behind lens Put object at focal point and image at infinityPut object at focal point and image at infinity Angular size of object is , bigger!Angular size of object is , bigger! Outgoing rays Rays seen coming from here  f f Image at Infinity  y

4. Angular Magnification (Standard) Without magnifier: 25 cm is closest distance to view Without magnifier: 25 cm is closest distance to view Defined by average near point. Younger people do betterDefined by average near point. Younger people do better   tan  = y / 25  tan  = y / 25 With magnifier: put object at distance p = f With magnifier: put object at distance p = f '  tan ' = y / f'  tan ' = y / f Define “angular magnification” m  = ' /  Define “angular magnification” m  = ' /  Note that magnifiers work better for older people because near point is actually > 25cm Note that magnifiers work better for older people because near point is actually > 25cm ~y/25 ’~y/f M  = ’/  = 25/f

5. Example Find angular magnification of lens with f = 5 cm Find angular magnification of lens with f = 5 cm

6. Eye Glasses Perfect Eye Nearsighted Nearsighted can be corrected with a diverging lens.  A far object can be focused on retina. Optical Instruments

7. A Farsighted Power of lens: diopter = 1/f (in m) (+) diopter  converging lens (-) diopter  diverging lens Larger diopter  Stronger lens (shorter f)

8. MaterialnCornea1.38 Aqueous Humor 1.33- 1.34 Lens 1.41- 1.45 Vitreous Humor 1.34 Air1.00 Water1.33

9. Combinations of Thin Lenses The image produced by the first lens is calculated as though the second lens were not present The image produced by the first lens is calculated as though the second lens were not present The light then approaches the second lens as if it had come from the image of the first lens The light then approaches the second lens as if it had come from the image of the first lens The image of the first lens is treated as the object of the second lens The image of the first lens is treated as the object of the second lens The image formed by the second lens is the final image of the system The image formed by the second lens is the final image of the system

10. Combination of Thin Lenses, 2 If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens p will be negativep will be negative The overall magnification is the product of the magnification of the separate lenses The overall magnification is the product of the magnification of the separate lenses

11. Combinations of Thin Lenses The image produced by the first lens is calculated as though the second lens were not present The image produced by the first lens is calculated as though the second lens were not present The light then approaches the second lens as if it had come from the image of the first lens The light then approaches the second lens as if it had come from the image of the first lens The image of the first lens is treated as the object of the second lens The image of the first lens is treated as the object of the second lens The image formed by the second lens is the final image of the system The image formed by the second lens is the final image of the system

12. Combination of Thin Lenses, 2 If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens p will be negativep will be negative The overall magnification is the product of the magnification of the separate lenses The overall magnification is the product of the magnification of the separate lenses

13. What is the combined focal length? 1/p 1 +1/q 1 =1/f 1 1/p 2 +1/q 2 =1/f 2 p 2 =-q 1 !!! 1/p 1 +1/q 2 =1/f 1 +1/f 2 1/f = 1/f 1 +1/f 2

14. Combination of Thin Lenses, example

15. Q. Two converging lenses are placed 20.0 cm apart. If the first lens has focal length of 10.0 cm and the second has a focal length of 20.0 cm, locate the. final image formed of a object 30.0 cm in front of the first lens. Find the magnification of the system 1/30 + 1/q = 1/10 q=+15 cm M 1 = -q/p = -0.5 For second lens: P= 20 –15 = 5 cm 1/5 +1/q = 1/20 q=-6.67 cm M 2 = -q/p = 1.33 M=M 1 M 2 =-0.667 Virtual, inverted

16. Compound Microscope (two converging lenses) objectiveeyepiece Real image formed by the objective lens  an object for the eyepiece lens Magnified inverted virtual image fofepopeqoqeMoMefofepopeqoqeMoMe popo qoqo pepe qeqe M = M o M e = (-q o /p o )(-q e /p e ) = (q o /p o )(q e /p e ) Each set follows lens Equations!!!

17. M = M o M e = (-q o /p o )(-q e /p e ) = (q o /p o )(q e /p e ) Magnification becomes larger q o >> p o : 1/p o + 1/q o = 1/f o 1/p o ≈ 1/f o The object should be put near the focal point of the object lens.

18. Q. In a compound microscope, the obj. lens has a focal length of 8 mm, And the eyepiece has a focal length 40 mm. The distance between the lenses is 200 mm. If the object is placed 8.4 mm from the obj. lens, What would be the magnification? M = (q o /p o )(q e /p e ) p o = 8.4 q o = ? f o = 8 D = 200 p e = ? q e = ? f e = 40 1/p o + 1/q o = 1/f o 1/q o = 1/f o – 1/p o = 1/8 – 1/8.4 q o = 168 p e = D – q e = 200 – 168 = 32 1/q e = 1/f e – 1/p e = 1/40 – 1/32 q e = -160 = (168/8.4)(-160/32) = -100 Inverted (M<0) but Virtual image (q e <0)

19. Telescope View Distant Objects View Distant Objects (Angular) Magnification M=f obj /f eye (Angular) Magnification M=f obj /f eye Increased Light Collection Increased Light Collection Large Telescope use Mirror Large Telescope use Mirror

21. Spherical Aberration Results from the focal points of light rays far from the principle axis are different from the focal points of rays passing near the axis Results from the focal points of light rays far from the principle axis are different from the focal points of rays passing near the axis For a mirror, parabolic shapes can be used to correct for spherical aberration For a mirror, parabolic shapes can be used to correct for spherical aberration

22. Chromatic Aberration Different wavelengths of light refracted by by a lens focus at different points Different wavelengths of light refracted by by a lens focus at different points Violet rays are refracted more than red raysViolet rays are refracted more than red rays The focal length for red light is greater than the focal length for violet lightThe focal length for red light is greater than the focal length for violet light Chromatic aberration can be minimized by the use of a combination of converging and diverging lenses Chromatic aberration can be minimized by the use of a combination of converging and diverging lenses

23. Multiple Lenses in Cameras Multiple lenses correct for various aberrations Multiple lenses correct for various aberrations Spherical aberration (poor focus at edge of lens)Spherical aberration (poor focus at edge of lens) Chromatic aberration (index of refraction varies with )Chromatic aberration (index of refraction varies with ) Gauss arrangement probably most commonGauss arrangement probably most common Actual arrangements are compromises! Actual arrangements are compromises! No perfect corrections for all factorsNo perfect corrections for all factors Balance of many factors, including costBalance of many factors, including cost