Chapter 34 onepage summary images images made by
Chapter 34: one-page summary images: • images made by mirrors – plane mirror i = +…: real i = –…: virtual – convex mirror – concave mirror m = +…: upright m = –…: inverted • images due to refraction • images made by lenses – diverging lens – converging lens See the convention used for this formula in the lecture notes; – lens maker’s formula It is different from the convention used in the textbook. • optical instruments – eye – corrective contact lenses and glasses – magnifying glass – telescope – microscope 1
Mirrors and conventions Mirror = a small piece of a sphere of radius R R=∞ R≠∞ flat concave: inner surface of a sphere convex: outer surface of a sphere Conventions: • focal distance f – flat: f =∞ – concave: f = +R/2 – convex: f = –R/2 • distance between object and mirror: p (>0 for single mirror) • distance between image and mirror: i = ± … (+ for real, – for virtual) – real image: light rays DO come from the point they seem to emerge from – virtual image: light rays DO NOT come from the point they appear to come from 2
How do we ”see” where objects are? 3
Image via reflection: flat mirror 4
Image via reflection: flat mirror 5
Image via reflection: flat mirror 6
Image via reflection: flat mirror 7
Image via reflection: flat mirror One sees an image at behind the mirror The image is virtual: rays only appear to come from 8
Image via reflection: flat mirror One sees an image at behind the mirror The image is virtual: rays only appear to come from 9
Image via reflection: flat mirror 10
Image via reflection: flat mirror Image is always: – virtual – upright – same size as the object 11
Ray rules for concave/convex mirrors O F V Mirror = small piece of a sphere of radius R V – vertex (center of the piece) O – center of a sphere F – focal point (R/2) çconcave: inner surface is used to reflect light convex: outer surface is used to reflect light Incident ray Reflected ray parallel to the mirror axis via/from the focal point F via/to the focal point F parallel to the mirror axis along radius at angle θ at mirror’s vertex V Using any pair of these rules allows one to find the image 12
Image via reflection: concave mirror (1) O V 13
Image via reflection: concave mirror (1) O V 14
Image via reflection: concave mirror (1) O V 15
Image via reflection: concave mirror (1) O V 16
Image via reflection: concave mirror (1) O V 17
Image via reflection: concave mirror (1) ratio of heights O V 18
Image via reflection: concave mirror (1) ratio of heights O V 19
Image via reflection: concave mirror (1) ratio of heights O V 20
Image via reflection: concave mirror (2) ratio of heights O V image is real: rays actually come from - by convention, in this case: i = x (positive) - image is inverted 21
Image via reflection: concave mirror (2) ratio of heights O V image is real: rays actually come from - by convention, in this case: i = x (positive) - image is inverted 22
Image via reflection: concave mirror (2) O V image is real: rays actually come from What if p<f ? - by convention, in this case: i = x (positive) - image is inverted i would come out negative – 23 what does it mean?
Image via reflection: concave mirror (3) O V 24
Image via reflection: concave mirror (3) O V image is virtual: rays only appear come from - by convention, in this case: i = –x (negative) - image is upright 25
Image via reflection: flat mirror Image: – can be real or virtual – can be inverted or upright – can be smaller or greater than the object https: //ophysics. com/l 10. html 26
Image via reflection: convex mirror O 27
Image via reflection: convex mirror O image is virtual: rays only appear come from - by convention, in this case: i = –x (negative) - image is upright 28
Image via reflection: flat mirror Image is always: – virtual – upright – smaller than the object https: //ophysics. com/l 10. html 29
HITT quiz Which of the three types of mirrors (flat, concave, convex) can form virtual images? (A) (B) (C) (D) (E) flat only concave only convex only flat and convex only all three 30
Images made by mirrors: SUMMARY flat concave convex ∞ +R/2 –R/2 image: real (i>0) or virtual (i<0)? virtual both possiblea virtual image: upright (m>0) or inverted (m<0) ? upright both possiblea upright image: shrunk (|m| < 1) or magnified (|m| >1) |m|=1 both possibleb shrunk Focal distance f a b distance of changeover: p = f distance of changeover: p = 2 f 31
Sample problem 32
Sample problem • 33
Sample problem 34
Sample problem 35
Image via refraction (1) At what depth does the bird see the fish? n The fish image seen by the bird is at a smaller depth! image is virtual: rays only appear to come from 36
Image via refraction (2) At what height does the fish see the bird? n The bird image seen by the fish is at a larger height! image is virtual: rays only appear to come from 37
Images by refraction (general expression) < n 2 n 1 r ( r <0 in this example ) 38
Lenses converging (convex) lens F focal distance f > 0 diverging (concave) lens F focal distance f < 0 39
Lens maker’s equation here, looking from outside, both surfaces are convex (R>0) Using the following convention: not the same as in the textbook!!! R > 0 for convex surfaces R < 0 for concave surfaces (regardless of where the observer is) index of refraction for outside medium (here, air/vacuum: n=1) here, looking from outside, both surfaces are concave (R<0) index of refraction for lens’s glass Regardless which side of a lens faces object. 40 Focal distance f fully describes the lens
Ray rules for converging lenses (1) V F Incident ray Refracted ray parallel to the lens axis via the focal point F parallel to the mirror axis thru lens center continue on Using any pair of these rules allows one to find the image 41
Ray rules for diverging lenses (2) V F Incident ray Refracted ray parallel to the lens axis from the focal point F toward the focal point F parallel to the mirror axis thru lens center continue on Using any pair of these rules allows one to find the image 42
Lenses • converging: • diverging: f>0 f<0 lens type focal distance (f) diverging converging image real (i>0) or virtual (i<0)? upright (m>0) or inverted (m<0)? magnified (|m|>1) or shrunk (|m|<1) ? – virtual upright shrunk + both possible(a) both possible(b) (a) (b) distance of changeover: p = f distance of changeover: p = 2 f 43
HITT quiz Which of the two types of lenses (converging, diverging) can form real images? (A) (B) (C) (D) (E) converging only diverging only both can form real images neither can form real images none of the above 44
Sample problem converging lens, focal distance = f object distance p = f/2 object size = h • Where is the image with respect to the lens? • Is it real or virtual? • Is it upright or inverted? • How large is it? 45
Sample problem converging lens, focal distance = f object distance p = f/2 object size = h • Where is the image with respect to the lens? • Is it real or virtual? • Is it upright or inverted? • How large is it? 46
Sample problem converging lens, focal distance = f object distance p = 2 f object size = h • Where is the image with respect to the lens? • Is it real or virtual? • Is it upright or inverted? • How large is it? 47
Sample problem converging lens, focal distance = f object distance p = 2 f object size = h • Where is the image with respect to the lens? • Is it real or virtual? • Is it upright or inverted? • How large is it? 48
Multiple lenses (e. g. 2): do one lens a time • LENS 1: – Find image 1 formed by the 1 st lens (ignore all others): 1/p 1 + 1/i 1 = 1/f 1 – This image is an object for the 2 nd lens • LENS 2 – If lens 2 does not obstruct formation of image 1 • find the distance x between the image and 2 nd lens • take p 2 = x (positive) • use 1/p 2 + 1/i 2 = 1/f 2 to find the image formed by the 2 nd lens – If lens 2 stands on the way of light rays forming image 1 • find the distance x between the would-be image and 2 nd lens • take p 2 = –x (negative) • use 1/p 2 + 1/i 2 = 1/f 2 to find the image formed by the 2 nd lens • Total magnification: 49
Sample problem Two lenses of focal length f are placed at x = 0 and x = 2 f. An object of size h is at x = – 0. 5 f. Find the position and size of the image formed by this set of two lenses. 50
Step 1: lens 1 Ignore second lens Find the image made by the first lens 51
Step 2: lens 2 (image made by lens 1 object for lens 2) Forget about the object and first lens Image made by the first lens is the object for second Find the image made by the second lens 52
Sample problem Two lenses of focal length f are placed at x = 0 and x = f. An object of size h is at x = – 2 f. Find the position and size of the image formed by this set of two lenses. 53
Solving graphically: Step 1: lens 1 Step 1: Ignore second lens. Find the image made by the first lens Oops! Second lens on a way of rays toward forming the image 54
Solving graphically: Step 1: lens 2 Oops! Second lens on a way of rays toward forming the image Luckily (in this particular), the red rays coming toward lens 2 can be easily refracted graphically 55
Solving using algebra: Step 1: lens 1 Step 1: Ignore second lens. Find the image made by the first lens 56
Solving using algebra: Step 2: oops! Oops! Second lens on a way of rays toward forming the image 57
Solving using algebra: Step 3: lens 2 Step 2: Ignore first lens. Find the image made by the 2 nd lens Second lens on a way of rays toward forming the image. The distance between the first would-be image and lens 2 should be treated as a negative p in the second step. 58
Camera • image: on a CCD matrix of pixels (used to be a film) • focus: for fixed f 0, adjust d=i for object at distance p • zoom: change f 0 (e. g. from 35 mm to 70 mm) – i will change (e. g. from ~35 mm to ~70 mm, for p >> f) – m=-i/p will change (e. g. by factor of 2 in this example) • diaphragm – amount of light – depth of field appearing sharp (in focus) • shutter speed – amount of light – freezing motion CCD matrix p 1 p 2 green point object green point on CCD red point object red disk on CCD R r i 1 i 2 59
Human eye • image: on retina • eyeball has fixed diameter: fixed i 0~24 mm • eye lens – converging – variable f is adjusted for desired p – average person: • far point: pmax = ∞ (relaxed eye, largest f: fmax = i 0 ) • near point: pmin = 25 cm (maximally strained eye, smallest f: fmin = … ~ 0. 9 fmax ) • diaphragm (opening in iris – pupil) – amount of light 60
HITT quiz Human’s eye lens is (A) diverging (B) converging (C) converging for near objects, diverging for far away objects (D) diverging for near objects, converging for far away objects (E) human eye does not have a lens 61
Glasses and contact lenses • Nearsighted person can see well near objects, but cannot see clearly far objects • Farsighted person can see well far objects, but cannot see clearly near objects • Correction lens of focal distance fcorr: for an object outside of the clear vision zone, the image made by the correction lens is placed into the clear vision zone • Diopter: , where fcorr is in meters 62
Lenses for nearsighted person • Farthest distance a person can see is short of infinity: dfar • Corrective lens makes a virtual image of an infinitely far away object at distance dfar away from an eye, where it can be comfortably seen 63
Lenses for farsighted person • Closest distance a person can see is dnear • Lens makes a virtual image of an object 25 cm away from an eye at distance dnear 64
Magnifying glass (f is small, << pmin) • Naked eye:
Magnifying glass (f is small, << pmin) • Naked eye: • With a magnifying glass (adjust p to get i = –pmin): The eye is just outside the lens
Microscope • Naked eye: • Microscope: –
Telescope (1/2) Naked eye and one star: B Naked eye and two stars: A 68
Telescope (2/2) Telescope makes an angle between stars larger by factor fobj/feye B A 69
Telescope (2/2) Telescope makes an angle between stars larger by factor fobj/feye B A Negative sign is added to signify that what you see such a telescope is inverted 70
- Slides: 70
