The last lecture Exam: Final: Consult the website, especially room assignments. Makeup: Register with me today. Tea and Cookies: Tuesdays 5PM, NPB 2175 A Question Unpolarized light of intensity I goes through an ideal polarizer. What is the intensity of the transmitted light? A: 3I / 2. B: I / 2. C: I / 4. D: Depends on θ. Simple Magnifier A simple magnifier consists of a single converging lens This device is used to increase the apparent size of an object The size of an image formed on the retina depends on the angle subtended by the eye Magnification by a Lens With a single lens, it is possible to achieve angular magnification up to about 4 without serious aberrations With multiple lenses, magnifications of up to about 20 can be achieved The multiple lenses can correct for aberrations 1
Compound Microscope A compound microscope consists of two lenses Gives greater magnification than a single lens The objective lens has a short focal length, ƒ o <1 cm The ocular lens (eyepiece) has a focal length, ƒ e, of a few cm Compound Microscope, cont The lenses are separated by a distance L L is much greater than either focal length The approach to analysis is the same as for any two lenses in a row The image formed by the first lens becomes the object for the second lens The image seen by the eye, I 2, is virtual, inverted and very much enlarged Magnifications of the Compound Microscope The lateral magnification of the microscope is ql L Ml = p ƒ l o The angular magnification of the eyepiece of the microscope is 25 cm m = e The overall magnification of the microscope is the product of the individual magnifications L 25 cm m = M m = l e ƒo ƒe ƒ e Other Considerations with a Microscope The ability of an optical microscope to view an object depends on the size of the object relative to the wavelength of the light used to observe it For example, you could not observe an atom (d 0.1 nm) with visible light (λ 500 nm) 2
Telescopes Mount Wilson Two fundamental types of telescopes Refracting telescope uses a combination of lenses to form an image Reflecting telescope uses a curved mirror and a lens to form an image Telescopes can be analyzed by considering them to be two optical elements in a row The image of the first element becomes the object of the second element The 100 inch (2.5 m) Hooker reflecting telescope at Mount Wilson Observatory near Los Angeles, California. http://en.wikipedia.org/wiki/gran_telescopio_canarias 3
http://hubble.nasa.gov/ http://www.jwst.nasa.gov/images_artist13532.html Refracting Telescope The two lenses are arranged so that the objective forms a real, inverted image of a distant object The image is near the focal point of the eyepiece The two lenses are separated by the distance ƒ o + ƒ e which corresponds to the length of the tube The eyepiece forms an enlarged, inverted image of the first image Angular Magnification of a Telescope The angular magnification depends on the focal lengths of the objective and eyepiece ƒo = = ƒ m θ θ o e Angular magnification is particularly important for observing nearby objects Very distant objects still appear as a small point of light 4
Problem 25.36 Galileo devised a simple terrestrial telescope that produces an upright image. It consists of a converging objective lens and a diverging eyepiece at opposite ends of the telescope tube. For distant objects, the tube length is the objective focal length less the absolute value of the eyepiece focal length. (a) Does the user of the telescope see a real or virtual image? (b) Where is the final image? (c) If a telescope is to be constructed with a tube 10.0 cm long and a magnification of 3, what are the focal lengths of the objective and eyepiece? Problem 25.36-II Note: We solve part (b) before answering part (a) in this problem. (b) The objective forms a real, diminished, inverted image of a very distant object at q 1 = f o. This image is a virtual object for the eyepiece at p e = - f e, giving q e -1 = 0 (a) Parallel rays emerge from the eyepiece, so the eye observes a virtual image. (c) The angular magnification is m = f o / f e = 3, giving f o = 3 f e. Also, the length of the telescope is L = f o +f e =3.00 f e f e =10.0 cm, giving f o = -5.00 cm and f e = 15 cm. Disadvantages of Refracting Telescopes Large diameters are needed to study distant objects Large lenses are difficult and expensive to manufacture The weight of large lenses leads to sagging which produces aberrations Reflecting Telescope Helps overcome some of the disadvantages of refracting telescopes Replaces the objective lens with a mirror The mirror is often parabolic to overcome spherical aberrations In addition, the light never passes through glass Except the eyepiece Reduced chromatic aberrations 5
Reflecting Telescope, Newtonian Focus The incoming rays are reflected from the mirror and converge toward point A At A, a photographic plate or other detector could be placed A small flat mirror, M, reflects the light toward an opening in the side and passes into an eyepiece Examples of Telescopes Reflecting Telescopes Largest in the world are 10 m diameter Keck telescopes on Mauna Kea in Hawaii Largest single mirror in US is 5 m diameter on Mount Palomar in California Refracting Telescopes Largest in the world is Yerkes Observatory in Wisconsin Has a 1 m diameter Resolution The ability of an optical system to distinguish between closely spaced objects is limited due to the wave nature of light If two sources of light are close together, they can be treated as noncoherent sources Because of diffraction, the images consist of bright central regions flanked by weaker bright and dark rings Rayleigh s Criterion If the two sources are separated so that their central maxima do not overlap, their images are said to be resolved The limiting condition for resolution is Rayleigh s Criterion When the central maximum of one image falls on the first minimum of another image, they images are said to be just resolved The images are just resolved when their angular separation satisfies Rayleigh s criterion 6
Just Resolved Barely Resolved (Left) and Not Resolved (Right) If viewed through a slit of width a, and applying Rayleigh s criterion, the limiting angle of resolution is λ θ min = a For the images to be resolved, the angle subtended by the two sources at the slit must be greater than θ min Resolution with Circular Apertures The diffraction pattern of a circular aperture consists of a central, circular bright region surrounded by progressively fainter rings The limiting angle of resolution depends on the diameter, D, of the aperture λ θ = 1.22 D min Resolving Power of a Diffraction Grating If λ 1 and λ 2 are nearly equal wavelengths between which the grating spectrometer can just barely distinguish, the resolving power, R, of the grating is λ λ R = λ λ λ 2 1 All the wavelengths are nearly the same 7
Resolving Power of a Diffraction Grating, cont A grating with a high resolving power can distinguish small differences in wavelength The resolving power increases with order number R = Nm N is the number of lines illuminated m is the order number All wavelengths are indistinguishable for the zeroth-order maximum m = 0 so R = 0 Another question Eye glasses for a nearsighted person require: A: converging lenses; B: diverging lenses; C: either ones depending on the type of nearsightedness. 8