LECTURE 25 Spherical Refracting Surfaces. Geometric Optics
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1 LECTURE 25 Spherical Refracting Surfaces Geometric ptics When length scales are >> than the light s wavelength, light propagates as rays incident ray reflected ray θ θ r θ 2 refracted ray Reflection: Refraction: θ i = θ r sinθ = sinθ 2 If > then θ 2 > θ If > then θ > θ 2 Refracted ray bends away from normal Refracted ray bends toward the normal
2 Spherically Refracting Surfaces Refracted ray directed toward the central axis A real image will form on that axis Refracted ray directed away from the central axis CANNT form a real image Backward extension of the ray can form a virtual image Real images Formed when object is relatively far from refracting surface Virtual images Formed when object is relatively near refracting surface Real C I r s s Virtual I C n 2 s r s Spherically Refracting Surfaces C I Real > Bends toward normal Real image formed s s r Real > Bends away from normal Real image formed C r I s s 2
3 Spherically Refracting Surfaces Virtual > I C Bends toward normal but away from central axis Virtual image formed > Bends away from normal and central axis Virtual Virtual image formed C I Refracting Surface Formula A point object is placed on the central axis of a convex refracting surface Center of curvature of the surface is at C Use small angle approximation If θ<< in radians, then sinθ θ. >% accuracy for θ<20 sinθ = sinθ 2 θ θ 2 θ = α + β, β = θ 2 + γ α + γ = ( )β α a c p, β = a c r, γ a c i s s p + = i s + s ' = n r 3
4 Sign Conventions for Refracting Surfaces bjects on incident-light side s is positive Images on refracted-light side s is positive Center of curvature on refracted-light side r is positive Parameters are negative if they don t meet above criteria for being positive Real C I r s s Summary: Spherically Refracting Surfaces. Real Images form on the side of a refracting surface that is opposite the object 2. Virtual Images form on the same side as the object 3. For light rays making only small angles with the central axis: s + s ' = n r 4. When the object faces a convex refracting surface, the radius of curvature is positive 5. When object faces a concave surface, radius of curvature is negative 4
5 Example: Mosquito in Amber A mosquito is embedded in amber with an index of refraction of.6. ne surface of the amber is spherically convex with a radius of curvature 3.0 mm. The mosquito head happens to be on the central axis of that surface, and when viewed along the axis appears to be buried 5.0 mm into the amber. How deep is it really? Draw a Picture Virtual I C Example: Mosquito in Amber s i = -5 mm bject and image are on same side of refracting surface (virtual image) =.6 Convention that object is in medium with index = (air) r = -3 mm Negative because the object faces a concave surface + = n s o s i r.6 + s o 5 =.6 3 s 0 = 4 mm 5
6 Refracting Surface: Example Consider =.0, =.8 R=0 cm s=5 cm M : Image is real, inverted and M >.0 5cm +.8 s.8 s =.8.0 0cm = ( )cm.8 s = = +35cm 0.333cm = 0.08cm s M = s = 35cm 5cm = 9 C object Definition Thin Lenses A lens is a transparent object with two refracting surfaces whose central axes coincide (= the central axis of the lens) Types Converging lens: causes rays initially parallel to the central axis to converge Diverging lens: a lens that causes rays initially parallel to the central axis to diverge Thin lens A lens whose thickest part is thin compared to the object distance, the image distance, and the radii of curvature of the two lens surfaces 6
7 Thin Lenses: Converging Lens C 2 F F 2 C r 2 f r Parallel rays refract twice Converge at F 2 a distance f from center of lens F 2 is a real focal point because rays pass through f > 0 for real focal points f = s Lens formula s ' f = (n ) Lens maker s r r 2 formula n > 0 since n glass > n air r > 0 r 2 < 0 since object facing convex surface since object facing concave surface Thin Lenses: Diverging Lens f < 0 for virtual focal points C F 2 F C 2 f Extension r r 2 Rays diverge, never pass through a common point F 2 at a distance f F 2 is virtual focal point 7
8 Thin Lenses A Double convex B Double concave Meniscus lens r r 2 /2 of A or B would be a plano-convex or plano-concave lens Demo Lens board converging diverging 8
9 Images from Thin Lenses C 2 F C I f r 2 r s s Convex lens bject outside focal point Image is real and inverted Images from Thin Lenses I F f s Convex lens bject inside focal point Image is virtual and same orientation as object s 9
10 Images from Thin Lenses C F 2 C 2 s r s r 2 Concave lens bject inside focal point Image is virtual and same orientation as object Properties of Images Converging Lenses (f>0) bject inside the focal point Image is virtual (i < 0), enlarged, has the same orientation, and farther from the lens bject outside the focal point Image is real (i > 0, meaning on the other side of the lens), inverted, reduced or enlarged (depending on the object distance), and closer to or farther from the lens. (like mirror) For magnification = (-), i = p = 2f (like mirror) bject at the focal point No image is formed: a (different) parallel bundle of rays is formed from each point of the object, and i (or s) = infinity 0
11 Properties of Images Diverging Lenses (f<0) bject inside or outside focal point Image is always virtual (i < 0), reduced, has the same orientation, and closer to the lens Special Rays for Refracting Surfaces A ray that is initially parallel to the central axis The refracted ray passes through the focal point A ray that passes through the focal point The refracted ray is parallel to the central axis A ray that passes through the center of curvature. The refracted ray propagates along the same direction as the incident ray. Works in either direction, of course
12 Locating Images by Drawing Rays F 3 2 F 2 I. Ray initially parallel to central axis will pass through F Ray passing through F will emerge parallel to the central axis. 3. Ray passing through center of lens will emerge with no change in direction because the ray encounters the two sides of the lens where they are almost parallel. Locating Images by Drawing Rays 2 F I F 2 3. Ray initially parallel to central axis will pass through F Backward extension of ray 2 passes through F 3. Ray 3 passes through center of lens will emerge with no change in direction. 2
13 Locating Images by Drawing Rays F 2 I F 2 3. Backward extension of ray passes through F 2 2. Extension of ray 2 passes through F 3. Ray 3 passes through center of lens will emerge with no change in direction Note: If image is located beyond lens 2, s 2 for lens 2 is negative Two Lens System Lens Lens 2 Let s represent distance from object,, to lens. Find s using: s = + f s s ' Ignore lens. Treat Image as for lens 2. Use s 2 = s. verall magnification: = + f 2 s 2 s 2 ' M = m m 2 = s ' s 2 ' s s 2 3
14 Example: Two Lens System Lens Lens 2 f f 2 A seed is placed in front of two thin symmetrical coaxial lenses (lens & lens 2) with focal lengths f =+24 cm & f 2 =+9.0 cm, with a lens separation of L=0.0 cm. The seed is 6.0 cm from lens. Where is the image of the seed? Lens : = + f s s ' s ' = 8.0cm s L Image is virtual Example: Two Lens System Lens Lens 2 f f 2 Lens 2: Treat image as 2 for lens 2. 2 is outside the focal point of lens 2. So, image 2 will be real & inverted on the other side of lens 2. s 2 = L + s ' = 8cm = + f 2 s 2 s 2 ' s L s 2 ' = 8.0cm Image 2 is real 4
15 Table for Lenses Lens Type bject Location Image Location Image Type Image rientation Sign of f, s, m Converging Inside F Same side as object Virtual Same as object +, -, + Converging utside F Side of lens opposite the object Real Inverted +, +, - Diverging Anywhere Same side as object Virtual Same as object -, -, + 5
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Thin Lenses f 4/16/2018 1 Thin Lenses: Converging Lens C 2 F 1 F 2 C 1 r 2 f r 1 Parallel rays refract twice Converge at F 2 a distance f from center of lens F 2 is a real focal pt because rays pass through
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