Chapter 33

(II) A lighted candle is placed 36 \(\mathrm{cm}\) in front of a converging lens of focal length \(f_{1}=13 \mathrm{cm},\) which in turn is 56 \(\mathrm{cm}\) in front of another converging lens of focal length \(f_{2}=16 \mathrm{cm}\) (see Fig. \(47 ) .(a)\) Draw a ray diagram and estimate the location and the relative size of the final image. (b) Calculate the position and relative size of the final image.

(I) A sharp image is located \(373 \mathrm{~mm}\) behind a \(215-\mathrm{mm}\) focal-length converging lens. Find the object distance (a) using a ray diagram, \((b)\) by calculation.

(I) Sunlight is observed to focus at a point \(18.5 \mathrm{~cm}\) behind a lens. (a) What kind of lens is it? (b) What is its power in diopters?

(1) Sunlight is observed to focus at a point 18.5 \(\mathrm{cm}\) behind a lens, \((a)\) What kind of lens is it? \((b)\) What is its power in dionters?

(I) (a) What is the power of a 23.5 -cm-focal-length lens? (b) What is the focal length of a -6.75 -D lens? Are these lenses converging or diverging?

(II) A certain lens focuses an object \(1.85 \mathrm{~m}\) away as an image \(48.3 \mathrm{~cm}\) on the other side of the lens. What type of lens is it and what is its focal length? Is the image real or virtual?

(II) A 105 -mm-focal-length lens is used to focus an image on the sensor of a camera. The maximum distance allowed between the lens and the sensor plane is \(132 \mathrm{~mm} .\) (a) How far ahead of the sensor should the lens be if the object to be photographed is \(10.0 \mathrm{~m}\) away? \((b) 3.0 \mathrm{~m}\) away? \((c) 1.0 \mathrm{~m}\) away? (d) What is the closest object this lens could photograph sharply?

(II) A stamp collector uses a converging lens with focal length \(28 \mathrm{~cm}\) to view a stamp \(18 \mathrm{~cm}\) in front of the lens. (a) Where is the image located? (b) What is the magnification?

(II) It is desired to magnify reading material by a factor of \(2.5 \times\) when a book is placed 9.0 \(\mathrm{cm}\) behind a lens. (a) Draw a ray diagram and describe the type of image this would be. (b) What type of lens is needed? (c) What is the power of the lens in diopters?

(II) A \(-8.00-\mathrm{D}\) lens is held \(12.5 \mathrm{~cm}\) from an ant \(1.00 \mathrm{~mm}\) high. Describe the position, type, and height of the image.

(II) \(\mathrm{A}-8.00\) -D lens is held 12.5 \(\mathrm{cm}\) from an ant 1.00 \(\mathrm{mm}\) high. Describe the position, type, and height of the image.

(II) An object is located \(1.50 \mathrm{~m}\) from an \(8.0-\mathrm{D}\) lens. By how much does the image move if the object is moved \((\) a) \(0.90 \mathrm{~m}\) closer to the lens, and ( \(b\) ) \(0.90 \mathrm{~m}\) farther from the lens?

(II) An object is located 1.50 \(\mathrm{m}\) from an 8.0 -D lens. By how much does the image move if the object is moved \((a) 0.90 \mathrm{m}\) closer to the lens, and \((b) 0.90 \mathrm{m}\) farther from the lens?

(II) (a) How far from a 50.0-mm-focal-length lens must an object be placed if its image is to be magnified \(2.50 \times\) and be real? \((b)\) What if the image is to be virtual and magnified \(2.50 \times ?\)

(II) How far from a converging lens with a focal length of \(25 \mathrm{~cm}\) should an object be placed to produce a real image which is the same size as the object?

(II) (a) A 2.80-cm-high insect is \(1.30 \mathrm{~m}\) from a \(135-\mathrm{mm}\) focal-length lens. Where is the image, how high is it, and what type is it? \((b)\) What if \(f=-135 \mathrm{~mm} ?\)

(II) How far from a converging lens with a focal length of 25 \(\mathrm{cm}\) should an object be placed to produce a real image which is the same size as the object?

(II) A bright object and a viewing screen are separated by a distance of \(86.0 \mathrm{~cm}\). At what location(s) between the object and the screen should a lens of focal length \(16.0 \mathrm{~cm}\) be placed in order to produce a sharp image on the screen? [Hint: first draw a diagram.

(II) How far apart are an object and an image formed by an \(85-\mathrm{cm}\) -focal-length converging lens if the image is \(2.95 \times\) larger than the object and is real?

(II) Show analytically that the image formed by a converging lens \((a)\) is real and inverted if the object is beyond the focal point \(\left(d_{0}>f\right),\) and \((b)\) is virtual and upright if the object is within the focal point \(\left(d_{\mathrm{o}}f,\) and \((d)\) for \(0<-d_{0}

(II) A converging lens has focal length \(f\). When an object is placed a distance \(d_{0}>f\) from this lens, a real image with magnification \(m\) is formed. \((a)\) Show that \(m=f /\left(f-d_{0}\right)\). (b) Sketch \(m\) vs, \(d_{\mathrm{o}}\) over the range \(f

(11) A converging lens has focal length \(f .\) When an object is placed a distance \(d_{0}>f\) from this lens, a real image with magnification \(m\) is formed. \((a)\) Show that \(m=f /\left(f-d_{0}\right)\) (b) Sketch \(m\) vs. \(d_{0}\) over the range \(f

(II) In a slide or movie projector, the film acts as the object whose image is projected on a screen (Fig. \(33-46\) ). If a 105 -mm-focallength lens is to project an image on a screen \(6.50 \mathrm{~m}\) away, how far from the lens should the slide be? If the slide is \(36 \mathrm{~mm}\) wide, how wide will the picture be on the screen?

(II) In a slide or movic projector, the film acts as the object whose image is projected on a screen (Fig, 46). If a 105 -mm-focal- length lens is to project an image on a screen 6.50 \(\mathrm{m}\) away.how far from the lens should the slide be? If the slide is 36 \(\mathrm{mm}\) wide, how wide will the picture be on the screen?

(III) A bright object is placed on one side of a converging lens of focal length \(f,\) and a white screen for viewing the image is on the opposite side. The distance \(d_{\mathrm{T}}=d_{\mathrm{i}}+d_{\mathrm{o}}\) between the object and the screen is kept fixed, but the lens can be moved. \((a)\) Show that if \(d_{\mathrm{T}}>4 f,\) there will be two positions where the lens can be placed and a sharp image will be produced on the screen. (b) If \(d_{\mathrm{T}}<4 f,\) show that there will be no lens position where a sharp image is formed. (c) Determine a formula for the distance between the two lens positions in part \((a),\) and the ratio of the image sizes.

(III) \((a)\) Show that the lens equation can be written in the Newtonian form: $$ x x^{\prime}=f^{2} $$ where \(x\) is the distance of the object from the focal point on the front side of the lens, and \(x^{\prime}\) is the distance of the image to the focal point on the other side of the lens Calculate the location of an image if the object is placed \(48.0 \mathrm{~cm}\) in front of a convex lens with a focal length of \(38.0 \mathrm{~cm}\) using \((b)\) the standard form of the thin lens formula, and \((c)\) the Newtonian form, derived above.

(II) A diverging lens with \(f=-33.5 \mathrm{~cm}\) is placed \(14.0 \mathrm{~cm}\) behind a converging lens with \(f=20.0 \mathrm{~cm} .\) Where will an object at infinity be focused?

(II) Two 25.0 -cm-focal-length converging lenses are placed \(16.5 \mathrm{~cm}\) apart. An object is placed \(35.0 \mathrm{~cm}\) in front of one lens. Where will the final image formed by the second lens be located? What is the total magnification?

(II) Two \(25.0-\mathrm{cm}\) -focal-length converging lenses are placed 16.5 \(\mathrm{cm}\) apart. An object is placed 35.0 \(\mathrm{cm}\) in front of one lens. Where will the final image formed by the second lens with a focal length of 38.0 \(\mathrm{cm}\) using \((b)\) the standard form of the thin lens formula, and \((c)\) the Newtonian form, derived above.

(II) A 34.0-cm-focal-length converging lens is \(24.0 \mathrm{~cm}\) behind a diverging lens. Parallel light strikes the diverging lens. After passing through the converging lens, the light is again parallel. What is the focal length of the diverging lens? [Hint: first draw a ray diagram.

(II) A diverging lens with a focal length of \(-14 \mathrm{~cm}\) is placed \(12 \mathrm{~cm}\) to the right of a converging lens with a focal length of \(18 \mathrm{~cm}\). An object is placed \(33 \mathrm{~cm}\) to the left of the converging lens. (a) Where will the final image be located? (b) Where will the image be if the diverging lens is \(38 \mathrm{~cm}\) from the converging lens?

(II) Two lenses, one converging with focal length \(20.0 \mathrm{~cm}\) and one diverging with focal length \(-10.0 \mathrm{~cm},\) are placed \(25.0 \mathrm{~cm}\) apart. An object is placed \(60.0 \mathrm{~cm}\) in front of the converging lens. Determine \((a)\) the position and \((b)\) the magnification of the final image formed. (c) Sketch a ray diagram for this system.

(II) A diverging lens is placed next to a converging lens of focal length \(f_{C},\) as in Fig. \(15 .\) If \(f\) represents the focal length of the combination, show that the focal length of the diverging lens, \(f_{\mathrm{D}},\) is given by \(\frac{1}{f_{\mathrm{D}}}=\frac{1}{f_{\mathrm{T}}}-\frac{1}{f_{\mathrm{C}}}\)

(II) A lighted candle is placed 36 \(\mathrm{cm}\) in front of a converging lens of focal length \(f_{1}=13 \mathrm{cm},\) which in turn is 56 \(\mathrm{cm}\) in front of another converging lens of focal length \(f_{2}=16 \mathrm{cm}\) (see Fig. \(47 ) .(a)\) Draw a ray diagram and estimate the location and the relative size of the final image. (b) Calculate the position and relative size of the final image.

(I) A double concave lens has surface radii of \(33.4 \mathrm{~cm}\) and \(28.8 \mathrm{~cm} .\) What is the focal length if \(n=1.58 ?\)

(I) Both surfaces of a double convex lens have radii of \(31.4 \mathrm{~cm} .\) If the focal length is \(28.9 \mathrm{~cm},\) what is the index of refraction of the lens material?

(II) An object is placed \(90.0 \mathrm{~cm}\) from a glass lens \((n=1.52)\) with one concave surface of radius \(22.0 \mathrm{~cm}\) and one convex surface of radius \(18.5 \mathrm{~cm}\). Where is the final image? What is the magnification?

(II) A prescription for a corrective lens calls for +3.50 diopters. The lensmaker grinds the lens from a "blank" with \(n=1.56\) and convex front surface of radius of curvature of \(30.0 \mathrm{~cm}\). What should be the radius of curvature of the other surface?

(I) A properly exposed photograph is taken at \(f / 16\) and \(\frac{1}{120}\) s. What lens opening is required if the shutter speed is \(\frac{1}{1000}\) s?

(I) A television camera lens has a 17 -cm focal length and a lens diameter of \(6.0 \mathrm{~cm}\). What is its \(f\) -number?

(II) Human vision normally covers an angle of about \(40^{\circ}\) horizontally. A "normal" camera lens then is defined as follows: When focused on a distant horizontal object which subtends an angle of \(40^{\circ}\), the lens produces an image that extends across the full horizontal extent of the camera's light-recording medium (film or electronic sensor). Determine the focal length \(f\) of the "normal" lens for the following types of cameras: \((a)\) a \(35-\mathrm{mm}\) camera that records images on film \(36 \mathrm{~mm}\) wide; \((b)\) a digital camera that records images on a charge-coupled device \((\mathrm{CCD}) 1.00 \mathrm{~cm}\) wide.

(II) A nature photographer wishes to photograph a \(38-\mathrm{m}\) tall tree from a distance of \(65 \mathrm{~m}\). What focal-length lens should be used if the image is to fill the 24 -mm height of the sensor?

(II) A person struggles to read by holding a book at arm's length, a distance of \(55 \mathrm{~cm}\) away. What power of reading glasses should be prescribed for her, assuming they will be placed \(2.0 \mathrm{~cm}\) from the eye and she wants to read at the "normal" near point of \(25 \mathrm{~cm} ?\)

(II) Reading glasses of what power are needed for a person whose near point is \(105 \mathrm{~cm}\), so that he can read a computer screen at \(55 \mathrm{~cm} ?\) Assume a lens-eye distance of \(1.8 \mathrm{~cm}\).

(II) An eye is corrected by a \(-4.50-\mathrm{D}\) lens, \(2.0 \mathrm{~cm}\) from the eye. \((a)\) Is this eye near- or farsighted? \((b)\) What is this eye's far point without glasses?

(II) A person's right eye can see objects clearly only if they are between \(25 \mathrm{~cm}\) and \(78 \mathrm{~cm}\) away. ( \(a\) ) What power of contact lens is required so that objects far away are sharp? (b) What will be the near point with the lens in place?

(II) A person has a far point of \(14 \mathrm{~cm}\). What power glasses would correct this vision if the glasses were placed \(2.0 \mathrm{~cm}\) from the eye? What power contact lenses, placed on the eye, would the person need?

(II) One lens of a nearsighted person's eyeglasses has a focal length of \(-23.0 \mathrm{cm}\) and the lens is 1.8 \(\mathrm{cm}\) from the eye If the person switches to contact lenses placed directly on the eye, what should be the focal length of the corresponding contact lens?

(II) What is the focal length of the eye lens system when viewing an object (a) at infinity, and (b) \(38 \mathrm{~cm}\) from the eye? Assume that the lens- retina distance is \(2.0 \mathrm{~cm}\).

(II) A nearsighted person has near and far points of 10.6 and \(20.0 \mathrm{~cm}\) respectively. If she puts on contact lenses with power \(P=-4.00 \mathrm{D}\), what are her new near and far points?