Physics 2C: Optics. refraction, Snell s law, polarization, images, thin mirrors, thin lenses July 11,

Transcription

1 Physics C: Optics Relection, reraction, Snell s law, polarization, images, thin mirrors, thin lenses July, 0 4

2 Relection: specularand diuse Size o objects a>>λ, treat waves as rays Light strikes medium, some transmitted, relected Angle o relectionangle o incidence When surace roughness >> λ, individual light rays ollow law o relection But... when roughness ~ λ...? (specular) θi θr specular diuse July, 0 5

3 Example: chapter 35 problem The mirrors in the igure make a 60 degree angle. A light ray enters parallel to the symmetry axis. (a) How many relectionsdoes it make? (b) Where and in what direction does it exit the mirror system? angle rom normal: # relections: 3 exits 80degrees rom initial direction July, 0 6

4 Reraction Snell s (Descartes ) Law Interace b/n (transparent) medium Relection Reraction Due to dierent speeds Frequency must be the same sinθivit/hypotenuse n sinθ/sinθ v/v index o reraction: nc/v Snell s law: nsinθnsinθ λvt θ θ λvt θ θ July, 0 7

5 Example Ch 35 Problem 4 A meter stick lies on the bottom o the rectangular trough with its zero mark at the let edge o the trough. You look into the long dimension o the trough at a 45 degree angle, with your line o sight just grazing the top edge o the tank. What mark on the meter stick do you see i the trough is (a) empty (b) hal ull o water (c) ull o water 40-cm Mirages Index o reraction o air is temperature dependent n increasing Light ollows curved path Several parallel layers continuously varying unction o requency a.) empty tan(45)l/(40cm) L(40cm)tan(45) L40 cm b.) ½ ull tan(45)l/(0cm) L0cm L(0cm)tanθ nsinθnsinθ sin45.33 sinθ Θ3., L3cm L33 cm c.) ull nsinθnsinθ sin45.33 sinθ Θ3., L(40cm)sin3. L6 cm July, 0 8 Caused by heated air due to sand

6 Total Internal Relection n<n, rays bent away rom normal (nsinθnsinθ) Critical angle: sinθ, sinθcn/n Above which 00% relected Example: Problem, 35-3: A compound lens is made rom crown glass (n.5) bonded to lint glass (n.89). What is the critical angle or light incident on the lint-crown interace? n.5 n.89 (n<n) sinθcn/n sinθc.5/.89 July, 0 9 sinθc53.5 degrees

7 Dispersion Index o reraction requency dependent Dierent wavelengths (colors) reracted through dierent angles (n increases with increasing requency) Basis or spectroscopy July, 0 0

8 Critical Angle and Rainbows Observer standing between sun and rain sees rainbow Max angle ~4degrees Index o reraction varies with λ, so does θmax Fainter, larger arc due to double internal relections (color order reversed) July, 0

9 90 - Polarizing (Brewster s) Angle Relection depends on interaction o E- ield component o EM wave with electrons (polariation) Recall directional antennas No EM radiation in direction o particle acceleration Polarization in plane o relection, 00% transmission i: Relected ray perpendicular to reracted ray Snell s law: nsinθbnsinθ θb+θ90, sinθcosθb tanθbn/n Unpolarized light relected polarized July, 0

10 d sinθ mλ a θ mλ Quiz 3 Formulas (Ch 37 & 35) S 4S λ mn nd ( m + λ )λ 0 πd sinθ cos λ sin φ ( ) φ φ sin ( ) θ r θ i sin S θ n c S 0 v n θ sin n sin θ min λ λ n n θ λ a θ c n n tan θ n B n July, 0 3 πa sinθ λ θ min.λ D (Quiz ) F ω π π k v T λ µ P I P µω A 4πr v β γp ρ v (Quiz ) B ρ ( I ) 0log I nλ L v v v E B S µ 0 E0B0 S µ 0 S S 0 cos θ p U P rad S c c 0 B P V V I I ± u ( P ) ρv v E cb I v v E E x c t 0 W / 0 m ω v λ k ρω s v sound 340m / s 8 c m / s 7 µ 4π 0 N A 0 ± displacement c 0 / / 0 v ( u v) Φ ε 0 t µ ε ε 8 C N m 0 0 v E

11 Plane Mirrors Mirror: image appears to be behind the mirror, upright, same size Consider the relection o as small object (point) Object appears as ar behind the mirror as it is in ront Observer perceives light rom image object image Image is virtual Smallest mirror to show ull image? Draw image relected, connect (red) rays rom observer s eye to top o head, bottom o eet, blue rays show actual path light travels Image reversed, L to R

12 yax^ Parabolic Mirror Light parallel to the symmetry axis will all relect through a special point called the FOCUS or FOCAL POINT Light rom ar away will converge at the ocus Point source at ocus will emerge as parallel

13 Spherical Mirror Similar to parabola near the apex Spheres are more regular so most mirrors are actually part o a spherical curve (spherical aberration) Use very small raction o the entire sphere Focal length >> size o mirror Light only strikes the mirror i they are close to and nearly parallel to the mirror axis (called paraxial rays)

14 Image Formation Need to ind or more rays rom each point on the object that converge to orm the image Any ray parallel to the mirror axis passes through ocus Any ray through ocus relects parallel to the axis Any ray through the center o the mirror relects symmetrically through the mirror axis Any ray through the center o curvature strikes the mirror normal to the mirror surace, and is relected 80 degrees Any rays is suicient to locate an image: o distance to object i distance to image ocal length o>c, C>o>, and C<

15 Image Formation (Concave) Spherical Mirrors object o C image i F object C image F o>c image is real, inverted, and reduced in size C>o> image is real, inverted, and magniied o< image is virtual, upright, and magniied object C F image

16 Concave vsconvex Mirrors Convex no possibility o real image, <0 Relected rays appear to diverge rom a common point behind the mirror Image is upright and reduced in size (used when image o a broad region needs to be captured in a small space)

17 Mirror Equation Consider the ray relected symmetrically about the mirror axis to derive magniication: Mh /h-i/o (M<0, inverted) Consider the ray relected through the ocus -h /h(i-)/i/o /i+/o/ I i<0, image is virtual I <0, mirror is convex Using ray thru C, C/ (R/) h o o h' Approximations true or C large compared with mirror size, and i i

18 Lenses (Thin, thickness<<radius o curvature) Transparent material that uses reraction to orm images Can be convex (parallel rays converge to a ocal point) or concave (parallel rays appear to diverge rom a ocal point) Thin lenses light bends just once Focal point on each side o lens Focal length same on both directions orientation doesn t matter Diverging lenses M<, can t use a diverging lens to start a ire!

19 Thin lens + object + ocus radius image object h (o) h (i)< 0 Rays parallel to lens axis reracted through ocus Any ray through the center o the lens undelected

20 Equation: again, similar triangles Essentially, the same triangles as or mirror object l ' h ' h h l image h < 0 l ' l + l l ' object h ' h h + o i h < 0

21 Lens Image Formation object h (o) h (i)< 0 o>, image is inverted and real o<, image is upright and virtual h (i)> 0 object h (o)

22 Lens Equation Same as that or mirrors! M-i/o (real images are inverted) /i+/0/ o>0 always i<0 image virtual (image on same side as lens), i>0 image real (not on same side as object) Diverging lenses: <0 Focal length Object distance Image distance Image + +, o> + Real, inverted + +, o< - Virtual, upright Virtual, upright o + i Find ocal length by changing iuntil the image upright

23 Reraction at a curved surace Light rays diverging rom point source are reracted to a common image point Restricted to small angles, Snell s nθnθ n(α+β)n(β -γ), treat AB as vertical α~ba/o, β~ba/r object γ~ba/i α n θ A True or any α, as long as small angle approximation is valid B o i radius R> 0 n n n n + this will become the o i R second surace o thin lens n θ β γ image

24 Reraction at a curved surace True or real, virtual images (i<0) I light incident rom opposite side, R<0 Works or lat suraces, R ininity Used or understanding thick lenses n A n α B γ image object Let hand ormula o n n n n + o i R i

25 Thick lenses interaces: image rom irst serves as the object o the other First: o n + i Second: Express in terms o thickness Lensmaker s ormula n R Ri<0 or diverging lenses Riininity or lat surace Conversion/diversion is reversed i surround medium has a higher index o reraction n + i o R Let hand ormula n o o + i t o n R ( n ) R R R i, 0, o t n i

26 Lens Aberrations Spherical (multiple oci) can be reduced by making incident rays parallel For close up objects, include only the central portion o the lens (camera) Chromatic due to index o reraction varying with wavelength (color) Reduced using composite lenses Unique to lenses

27 Multiple Mirrors Single mirror 90 degrees 60 degrees θ? (ininite)

28 Curved Mirrors /i+/o/ Magniication M-i/o i>0 real, inverted i<0 virtual, upright Concave (>0) i>0 i o> ( M < or o>r) i<0 i o< ( M >) Convex (<0) i<0 ONLY M < only Virtual images appear to be behind the mirror, while light rom a real image cannot be distinguished rom the object

29 Thin Lenses /o+/i/ M-i/o Convex (converging) >0 o>, i>0 (real, inverted) o> M < o<, i<0 (virtual, upright) M > Concave (diverging) <0 i<0 (virtual, upright) M < Virtual images can only be seen through the lens, while light actually emanates rom the real image location

30 (Thick Lenses) Contact lenses + i o (R<0 i concave to object) R > 0 R ~ > 0 R ~ R < 0 > 0 Thicker on the edge than in the center concave (diverging) ( n ) R R Thicker in the center than on the edges -convex (converging) R < n> R n> R > R < 0 R < R > 0

31 Simple Optical Instruments Film projector >0, i>0, M> (<o<) Magniying glass >0, i i<0, M > I <0, M < Side-view mirrors <0, M < (objects may appear closer )

32 The Eye Light passes through lens (adjustable ocal length) Focused, real image ormed on retina Myopic (near-sighted): image orms in ront o retina (concave lenses) Hyperopic (ar-sighted): image orms behind the retina (convex lenses) Near point: min distance below which the eye cannot ocus sharply (~5cm) corrective power (diopters)/

33 Eyeglasses Reading glasses: can t ocus closer than x cm? Want lenses with ocal length to produce virtual x cm or an object located at the standard near point, 5 cm + o i + 5 x Since x>5cm, >0 (converging lenses) Make sure that lens power is /, where is in meters Nearsighted person cannot see clearlybeyond y cm? Prescribelens power produce virtual y cm or objects very ar away + y y

34 Camera Similar to eye image ocused on ilm (adjustable lens) Adjustable diaphragm (reduces spherical aberrations) Broader range o distances in ocus Zoom lenses: adjust ocal length (magniication o the same subject) Ch 36 Problem 5 Camera s zoom lens covers ocal length range rom 38 mm to 0 mm. You photograph an object irst with 38mm ocal length then with 0mm ocal length. Compare the size o the object. M i o Same object position + + o i o i M i o o o o i i

35 Simple magniier (converging lens) You can get the object closer, and still ocus on it Angular magniication (m) must be deined relative to some observing distance Maximum m w/image at near point Image ormed ar object + o ~ o virtual image (absolute magniication is irrelevant) angular size, α 5 cm Angle subtended by object in lens ar ( angular) m object o~ e β α naked eye Angle subtended by object in lens 5 cm e β eyepiece lens (close)

36 object h Compound Microscope Single lens: spherical aberrations m>4 l o Object placed just beyond ocal length o objective inal virtual image (absolute magniication is irrelevant) objective lens (Small o) Lens spacing L (Magniied, real image) h ' L M h (e<<l) o slightly intermediate < e image Image just inside ocal length o eyepiece ( angular) m eyepiece lens (Simple magniier) 5 cm e Total angular: m total Mm As long as object size >> λ(geometric requirement) L o 5 cm e

37 Telescope Only angular magniication, m, is meaningul Overall magniication ratio o m s object h? Reracting or relecting Images distant objects at ocal length l,000,000 light years inal virtual image (absolute magniication is irrelevant) objectiv e lens α α h ' o o intermediat e image slightly < e β h ' β e eyepiece lens β o ( angular) m α e