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Multiple Lenses Draw Rays ! • We determine the effect of a system of lenses by considering the image of one lens to be the object for the next lens. +1 0 -1 +2 ∴ s1 = +1.5m, f1 = +1m s = 3m ' 1 For the second lens: ∴ s2' = −0.8 m +4 +6 +5 f = -4m f = +1m For the first lens: +3 s1' m1 = − = − 2 s1 s2 = +1m, f2 = -4m s 2' 4 m2 = − =+ s2 5 1 1 1 1 1 1 = − = − = ' s1 f1 s1 1 m 1.5 m 3 m 1 1 1 1 1 5 = − = − = − 4m s2' f 2 s2 − 4 m 1 m m = m 1m 2 = − 8 5 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): +1 0 -1 +2 +3 +4 +5 +6 f = +1m For the first lens: ∴ f = -4m s1 = +1.5m, f1 = +1m s = 3m ' 1 For the second lens: ∴ s2' = +4 m s1' m1 = − = − 2 s1 s2 = -2m, f2 = -4m 1 1 1 1 1 1 = − = − = s1' f1 s1 1 m 1.5 m 3 m 1 1 1 1 1 1 = − = − = s2' f 2 s2 − 4 m − 2 m 4 m s 2' m2 = − = +2 s2 Note the negative object distance for the 2nd lens. m = m1m2 = −4 The eye • The “Normal Eye” The 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 1 1 1 1 = + ' = 0+ f s s 2.5 cm f = 2.5 cm 1 1 1 1 1 = + ' = + f s s 25 2.5 f = 2.3 cm 2.5cm 25cm 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! This is called “accommodation” Diopter: 1/f Eye = 40 diopters, accommodates by about 10%, or 4 diopters Eye corrections (glasses, contacts) Near-sighted eye is elongated, image of distant object forms in front of retina Add diverging lens, image forms on retina Far-sighted eye is short, image of close object forms behind retina power = 1/f; f in meters Add converging lens, image forms on retina The farsighted eye Near point changes with age: 7 cm → 200 cm 7 years 60 years Most distinct vision is at near point. Image is largest. Example: Reading glasses. The near point of a person’s eye is 75 cm. What power reading glasses should be used to bring the near point to 25 cm? s = 25 cm s ′ = −75 cm 1 1 1 1 1 + = = + s s ′ f 25 cm − 75 cm 1 1 = = 2.67 diopters f 0.375 m i virtual F o s’ F s Assumes eye very close. Results are slightly different when distance between them is taken ino consideration. Old Preflight A person with normal vision (near point 28cm) is standing in front of a plane mirror. What is the closest distance to the mirror the person can stand and still see himself in focus? a) 14 cm b) 28 cm c) 56 cm s s’ Object for eye = Image of self • Distance from eye to object = s+s’ • Set s+s’ = near point Old Preflight Two people who wear glasses are camping. One of them is nearsighted and the other is farsighted. Which person's glasses will be useful in starting a fire with the sun's rays? a) the farsighted person's glasses b) the nearsighted person's glasses What do you need to start a fire? • REAL IMAGE ! (light is focused to a point) • Converging lens gives REAL IMAGE • Far-sighted people need converging lenses! Special Lens Combinations If two thin lenses are close together, they act effectively as a single lens. The focal length of the “doublet” is given by 1 f doublet 1 1 = + f1 f2 f1 f2 fdoublet Note the power (=1/f) of the combination is just Pdoublet = P1 + P2 Lecture 27, ACT 2 • Hildegard’s retina is 2.5 cm behind the lens, which has a minimum focal length of 2.6 cm. 2.5 cm feye = 2.6 cm 1. What does the focal length fcl of her contact lens need to be? (a) 65 cm (b) -65 cm (c) -0.1 cm 2. What is the power Pcl of the contact lens? (a) 1.5 D (b) -1.5 D (c) 1000 D Lecture 27, ACT 2 • Hildegard’s retina is 2.5 cm behind the lens, which has a minimum focal length of 2.6 cm. 2.5 cm feye = 2.6 cm 1. What does the focal length fcl of her contact lens need to be? (a) 65 cm 1 1 1 = + f f eye f cl (b) -65 cm (c) -0.1 cm 1 1 1 = − f cl 2.5 cm f eye f eye (2.5 cm ) 2.6 × 2.5 f cl = = = 65 cm f eye − 2.5 cm 2.6 − 2.5 Lecture 27, ACT 2 • Hildegard’s retina is 2.5 cm behind 2.5 cm the lens, which has a minimum focal length of 2.6 cm. f = 2.6 cm 2. What is the power Pcl of the contact lens? eye (a) 1.5 D (b) -1.5 D (c) 1000 D 1 1 Pcl ≡ = = 1.5 D f cl 0.65 m Note: We could have solved for the power directly: Pcl ≡ Pneed − Peye = 40 D − 38.5 D Angular Magnification • Our sense of the size of an object (in the absence of other clues) is determined by the size of image on the retina. • This is proportional to the angle subtended by the object: h α1 Bigger image h’1 ~ α1 h α2 h’2 ~ α2 Smaller image • The magnification of an optical system can then be equivalently defined as the ratio of the output angular spread to the input angular spread: M ≡ θ out θ in Magnifying Glass • Our sense of the size of an object is determined by the size of image on the retina. – If the object were closer to our eye, it would subtend a larger angle. – However, we can only focus on an object if it is no closer than the near-point distance Lnp (~25 cm). – We can use a simple magnifier to create an enlarged virtual image outside Lnp. Positive “f” lens Lnp h α h β ~f Object at Near Point - can’t get nearer h α ≈ L np Define Angular Magnification: β Lnp M ≡ ≈ α f Object just inside Focal Point of simple magnifier h β≈ f ⇒ Choose f < Lnp Ma gnifier Ma At normal distances a small object (small letter print) subtends a small angle θ. with a lens or magnifier we can magnify the subtended angle to θ′. The ratio is call angular magnification θ′ M = θ Which is NOT the same as lateral magnification. Telescopes • The purpose of a telescope is to gather light from distant objects and produce a magnified image. – Refracting telescopes use lenses so that the objects can be viewed directly. θ1 f1 θ1 ≈ f2 θ2 θ1 M = h’ h' f1 θ2 ≈ h' f2 θ2 h '/ f 2 f = = 1 h '/ f 1 f2 θ1 – Reflecting telescopes use mirrors to create the image » Most astronomical telescopes are reflectors, since the most important feature for these telescopes is the light gathering ability, and it is easier to make a large mirror than it is to make large lenses. detector Telescope Angular magnification y′ / θ′ M = =− θ y/ f2 =− f1 f1 f2 Note, light gathering power depends on objective lens Diameter. Hubble Space Telescope The HST was launched in 1990; it was discovered that a lens had been ground incorrectly, so all images were blurry! A replacement “contact lens”, COSTAR, was installed in 1993. (Now, Hubble’s instruments have built-in corrective optics, so COSTAR is no longer needed.) Before COSTAR After COSTAR Aperture of primary mirror: 2.4 m (~8 ft.) Mass of primary mirror: 828 kg (~1800 lbs)