2. Physics 1402: Lecture 31 Today’s Agenda Announcements: –Midterm 2: Monday Nov. 16 … –Homework 08: due Wednesday (after midterm 2) Optics –Lenses –Eye

3. o i f h’ h R   h o-R R-i o i &

4. Summary We have derived, in the paraxial (and thin lens) approximation, the same equations for mirrors and lenses: when the following sign conventions are used: Variable f > 0 f < 0 o > 0 o < 0 i > 0 i < 0 Mirror concave convex real (front) virtual (back) real (front) virtual (back) Lens converging diverging real (front) virtual (back) real (back) virtual (front)

5. This could be used as a projector. Small slide on big screen This is a magnifying glass This could be used in a camera. Big object on small film Upright Enlarged Virtual Inverted Enlarged Real Inverted Reduced Real ImageObject Inside F Object Image Past 2F Image Object Between F & 2F 3 Cases for Converging Lenses

6. 1) Rays parallel to principal axis pass through focal point. 2) Rays through center of lens are not refracted. 3) Rays toward F emerge parallel to principal axis. F F Object P.A. Image is virtual, upright and reduced. Image Diverging Lens Principal Rays

7. Multiple Lenses We determine the effect of a system of lenses by considering the image of one lens to be the object for the next lens. For the first lens: o 1 = +1.5, f 1 = +1 For the second lens: o 2 = +1, f 2 = -4   f = +1 f = -4 +3 +1 0 +2+6 +5+4

8. Multiple Lenses Objects of the second lens can be virtual. Let’s move the second lens closer to the first lens (in fact, to its focus): For the first lens: o 1 = +1.5, f 1 = +1 For the second lens: o 2 = -2, f 2 = -4   Note the negative object distance for the 2nd lens. f = +1 f = -4 +3 +1 0 +2+6 +5+4

9. Multiple Lenses If the two lenses are thin, they can be touching – i.e. in the same position. We can treat as one lens. f total = ?? ? Adding, For the first lens: o=o 1, i 1 and f 1 For the second lens: o 2 = -i 1, i 2 =i, f 2 As long as,

12. The Lens Equation –Convergent Lens: i f h’ o h

13. The Lensmaker’s Formula So far, we have treated lenses in terms of their focal lengths. How do you make a lens with focal length f ? Start with Snell’s Law. Consider a plano-convex lens: Snell’s Law at the curved surface: The bend-angle  is just given by: The bend-angle  also defines the focal length f: The angle  can be written in terms of R, the radius of curvature of the lens : Putting these last equations together, R N air h     light ray Assuming small angles,

14. More generally…Lensmaker’s Formula Two curved surfaces… Two arbitrary indices of refraction R > 0 if convex when light hits it R < 0 if concave when light hits it The complete generalized case… Note: for one surface Planar,

15. ~f e I1I1 eyepiece I2I2 ~f o objective L The EYE

16. Retina To brain The Eye What does the eye consist of? –Sphere (balloon) of water. - An aperture that controls how much light gets through – the Iris/pupil - Bulge at the front – the cornea - A variable focus lens behind the retina – the lens - A screen that is hooked up to your brain – the retina Cornea Iris Lens

17. The Eye The “Normal Eye” –Far Point  distance that relaxed eye can focus onto retina =  –Near Point  closest distance that can be focused on to the retina = 25 cm Therefore the normal eye acts as a lens with a focal length which can vary from 2.5 cm (the eye diameter) to 2.3 cm which allows objects from 25 cm   to be focused on the retina! 2.5cm 25cm this is called “accommodation” Diopter: 1/f Eye = 40 diopters, accommodates by about 10%, or 4 diopters

18. Lecture 31, ACT 1 When your eye adjusts to read versus see far objects, its muscles adjust so that the lens bulges and elongates. To read a book do we want a bulged lens or an elongated lens ?

19. Cornea Lens F < D F N < F F Near Case We have f 1 = f cornea, f 2 = f lens For F to get smaller, so must f lens Smaller f means more curvature (see lensmakers formula) Cornea Lens D = F F Far Away Case D Bonus: Calculate how much the radius of curvature of the lens changes as the eye adjusts from the far to the near point. Now since,

20. Getting Old As you age, the lens loses its ability to change its shape. It gets stuck in its relaxed position, the far point. Thus the eye is now just an unadjustable lens. Objects at different distances will focus at different places. Only objects at infinity will focus on the retina. 2.5cm 25cm This is called presbyopia, it is not necessarily “farsightedness”.

21. An intuitive way to view eye corrections Near-sighted eye is elongated, image forms in front of retina Add diverging lens, image forms on retina Far-sighted eye is short, image forms behind retina Add converging lens, image forms on retina Note: for old age (presbyopia), this sort of correction can only make one point in focus. If your relaxed eye naturally focuses either at infinity (for driving) or the near point (reading) then you only need one lens. Otherwise bifocals are needed. Could you design multifocals ??

24. Magnification Our sense of the size of an object is determined by the size of image on the retina. –Consequently, the relevant magnification factor of a lens is just the ratio of the angular size with the lens to the angular size without the lens.  L np h Object at Near Point ~f  h Object just inside Focal Point of simple magnifier Define Angular Magnification:

26. Compound Microscope o1o1 h O I2I2 h2h2 f eye h1h1 I1I1 i1i1 Objective (f ob < 1cm) f ob L Eyepiece (f eye ~5cm) Magnification:

27. Refracting Telescope Star f eye I2I2 h2h2 f ob Objective (f ob ~ 250cm) Eyepiece (f eye ~5cm) i1i1 I1I1 h1h1 Angular Magnification:    