Chapter 33

(II) The closely packed cones in the fovea of the eye have a diameter of about \(2 \mu \mathrm{m}\). For the eye to discern two images on the fovea as distinct, assume that the images must be separated by at least one cone that is not excited. If these images are of two point-like objects at the eye's \(25-\mathrm{cm}\) near point, how far apart are these barely resolvable objects? Assume the diameter of the eye (cornea-to-fovea distance) is \(2.0 \mathrm{~cm} .\)

(I) A magnifier is rated at \(3.0 \times\) for a normal eye focusing on an image at the near point. (a) What is its focal length? (b) What is its focal length if the \(3.0 \times\) refers to a relaxed eye?

(II) Sherlock Holmes is using an 8.80 -cm-focal-length lens as his magnifying glass. To obtain maximum magnification, where must the object be placed (assume a normal eye), and what will be the magnification?

(II) A small insect is placed \(5.85 \mathrm{~cm}\) from a +6.00 -cm-focallength lens. Calculate ( \(a\) ) the position of the image, and (b) the angular magnification.

(II) A 3.40-mm-wide bolt is viewed with a \(9.60-\mathrm{cm}\) -focallength lens. A normal eye views the image at its near point. Calculate \((a)\) the angular magnification, \((b)\) the width of the image, and (c) the object distance from the lens.

(II) A magnifying glass with a focal length of \(9.5 \mathrm{~cm}\) is used to read print placed at a distance of \(8.3 \mathrm{~cm} .\) Calculate \((a)\) the position of the image; \((b)\) the angular magnification.

(II) A magnifying glass is rated at \(3.0 \times\) for a normal eye that is relaxed. What would be the magnification for a relaxed eye whose near point is (a) \(65 \mathrm{~cm},\) and (b) \(17 \mathrm{~cm} ?\) Explain the differences.

(II) A converging lens of focal length \(f=12 \mathrm{~cm}\) is being used by a writer as a magnifying glass to read some fine print on his book contract. Initially, the writer holds the lens above the fine print so that its image is at infinity. To get a better look, he then moves the lens so that the image is at his \(25-\mathrm{cm}\) near point. How far, and in what direction (toward or away from the fine print) did the writer move the lens? Assume the writer's eye is adjusted to remain always very near the magnifying glass.

(I) What is the magnification of an astronomical telescope whose objective lens has a focal length of \(78 \mathrm{~cm}\), and whose eyepiece has a focal length of \(2.8 \mathrm{~cm} ?\) What is the overall length of the telescope when adjusted for a relaxed eye?

(I) The overall magnification of an astronomical telescope is desired to be \(35 \times\). If an objective of \(88 \mathrm{~cm}\) focal length is used, what must be the focal length of the eyepiece? What is the overall length of the telescope when adjusted for use by the relaxed eye?(I) The overall magnification of an astronomical telescope is desired to be \(35 \times\). If an objective of \(88 \mathrm{~cm}\) focal length is used, what must be the focal length of the eyepiece? What is the overall length of the telescope when adjusted for use by the relaxed eye?

(1) The overall magnification of an astronomical telescope is desired to be \(35 \times .\) If an objective of 88 \(\mathrm{cm}\) focal length is used, what must be the focal length of the eyepiece? What is the overall length of the telescope when adjusted for use by the relaxed eye?

(II) A \(7.0 \times\) binocular has 3.0 -cm-focal-length eyepieces What is the focal length of the objective lenses?

(II) An astronomical telescope has an objective with focal length \(75 \mathrm{~cm}\) and a \(+35 \mathrm{D}\) eyepiece. What is the total magnification?

(II) An astronomical telescope has its two lenses spaced \(78.0 \mathrm{~cm}\) apart. If the objective lens has a focal length of \(75.5 \mathrm{~cm},\) what is the magnification of this telescope? Assume a relaxed cye.

(II) A Galilean telescope adjusted for a relaxed eye is \(33.8 \mathrm{~cm}\) long. If the objective lens has a focal length of \(36.0 \mathrm{~cm},\) what is the magnification?

(II) What is the magnifying power of an astronomical telescope using a reflecting mirror whose radius of curvature is \(6.4 \mathrm{~m}\) and an eyepiece whose focal length is \(2.8 \mathrm{~cm} ?\)

(II) The Moon's image appears to be magnified \(120 \times\) by a reflecting astronomical telescope with an eyepiece having a focal length of 3.1 \(\mathrm{cm} .\) What are the focal length and radius a of curvature of the main (objective) mirror?

(II) A \(120 \times\) astronomical telescope is adjusted for a relaxed eye when the two lenses are \(1.25 \mathrm{~m}\) apart. What is the focal length of each lens?

(II) \(\mathrm{A} 120 \times\) astronomical telescope is adjusted for a relaxed eye when the two lenses are 1.25 \(\mathrm{m}\) apart. What is the focal length of each lens?

(II) An astronomical telescope longer than about \(50 \mathrm{~cm}\) is not easy to hold by hand. Based on this fact, estimate the maximum angular magnification achievable for a telescope designed to be handheld. Assume its eyepiece lens, if used as a magnifying glass, provides a magnification of \(5 \times\) for a relaxed eye with near point \(N=25 \mathrm{~cm}\).

(I) A microscope uses an eyepiece with a focal length of \(1.50 \mathrm{~cm}\). Using a normal eye with a final image at infinity, the barrel length is \(17.5 \mathrm{~cm}\) and the focal length of the objective lens is \(0.65 \mathrm{~cm}\). What is the magnification of the microscope?

(I) A \(680 \times\) microscope uses a \(0.40-\mathrm{cm}\) -focal-length objective lens. If the barrel length is \(17.5 \mathrm{~cm},\) what is the focal length of the eyepiece? Assume a normal eye and that the final image is at infinity.

(I) A \(680 \times\) microscope uses a 0.40 -cm-focal-length objective lens. If the barrel length is \(17.5 \mathrm{cm},\) what is the focal length of the eyepiece? Assume a normal eye and that the final image is at infinity.

(I) A 17 -cm-long microscope has an eyepiece with a focal length of \(2.5 \mathrm{~cm}\) and an objective with a focal length of \(0.28 \mathrm{~cm} .\) What is the approximate magnification?

(II) A microscope has a \(13.0 \times\) eyepiece and a \(58.0 \times\) objective lens \(20.0 \mathrm{~cm}\) apart. Calculate ( \(a\) ) the total magnification, (b) the focal length of each lens, and (c) where the object must be for a normal relaxed eye to see it in focus.

(II) A microscope has a 1.8 -cm-focal-length eyepiece and a \(0.80-\mathrm{cm}\) objective. Assuming a relaxed normal eye, calculate (a) the position of the object if the distance between the lenses is \(16.8 \mathrm{~cm},\) and \((b)\) the total magnification.

(II) The eyepiece of a compound microscope has a focal length of \(2.80 \mathrm{~cm}\) and the objective lens has \(f=0.740 \mathrm{~cm} .\) If an object is placed \(0.790 \mathrm{~cm}\) from the objective lens, calculate (a) the distance between the lenses when the microscope is adjusted for a relaxed eye, and ( \(b\) ) the total magnification.

(II) The eyepiece of a compound microscope has a total length of 2.80 \(\mathrm{cm}\) and the objective lens has \(f=0.740 \mathrm{cm} .\) I an object is placed 0.790 \(\mathrm{cm}\) from the objective lens, calculate (a) the distance between the lenses when the microscopeiadjusted for a relaxed eve, and \((b)\) the total magnification.

(II) An inexpensive instructional lab microscope allows the user to select its objective lens to have a focal length of \(32 \mathrm{~mm}, 15 \mathrm{~mm},\) or \(3.9 \mathrm{~mm} .\) It also has two possible eyepieces with magnifications \(5 \times\) and \(10 \times\). Each objective forms a real image \(160 \mathrm{~mm}\) beyond its focal point. What are the largest and smallest overall magnifications obtainable with this instrument?

(III) Given two 12 -cm-focal-length lenses, you attempt to make a crude microscope using them. While holding these lenses a distance \(55 \mathrm{~cm}\) apart, you position your microscope so that its objective lens is distance \(d_{0}\) from a small object. Assume your eye's near point \(N=25 \mathrm{~cm}\). (a) For your microscope to function properly, what should \(d_{\mathrm{o}}\) be? (b) Assuming your eye is relaxed when using it, what magnification \(M\) does your microscope achieve? (c) Since the length of your microscope is not much greater than the focal lengths of its lenses, the approximation \(M \approx N \ell / f_{\mathrm{c}} f_{\mathrm{o}}\) is not valid. If you apply this approximation to your microscope, what \% error do you make in your microscope's true magnification?

(II) A planoconvex lens (Fig. 33-2a) has one flat surface and the other has \(R=15.3 \mathrm{~cm} .\) This lens is used to view a red and yellow object which is \(66.0 \mathrm{~cm}\) away from the lens. The index of refraction of the glass is 1.5106 for red light and 1.5226 for yellow light. What are the locations of the red and yellow images formed by the lens?

(II) An achromatic lens is made of two very thin lenses, placed in contact, that have focal lengths \(f_{1}=-28 \mathrm{~cm}\) and \(f_{2}=+25 \mathrm{~cm} .\) (a) Is the combination converging or diverging? (b) What is the net focal length?

A \(200-\mathrm{mm}\) -focal-length lens can be adjusted so that it is \(200.0 \mathrm{~mm}\) to \(206.4 \mathrm{~mm}\) from the film. For what range of object distances can it be adjusted?

A 200 -mm-focal-length lens can be adjusted so that it is 200.0 \(\mathrm{mm}\) to 206.4 \(\mathrm{mm}\) from the film. For what range of object distances can it be adjusted?

If a 135-mm telephoto lens is designed to cover object distances from \(1.30 \mathrm{~m}\) to \(\infty\), over what distance must the lens move relative to the plane of the sensor or film?

For a camera equipped with a 58 -mm-focal-length lens, what is the object distance if the image height equals the object height? How far is the object from the image on the film?

Show that for objects very far away (assume infinity), the magnification of any camera lens is proportional to its focal length.

A converging lens with focal length of \(13.0 \mathrm{~cm}\) is placed in contact with a diverging lens with a focal length of \(-20.0 \mathrm{~cm} .\) What is the focal length of the combination, and is the combination converging or diverging?

An astronomical telescope has a magnification of \(8.0 \times .\) If the two lenses are \(28 \mathrm{~cm}\) apart, determine the focal length of each lens.

(a) Show that if two thin lenses of focal lengths \(f_{1}\) and \(f_{2}\) are placed in contact with each other, the focal length of the combination is given by \(f_{\mathrm{T}}=f_{1} f_{2} /\left(f_{1}+f_{2}\right)\). (b) Show that the power \(P\) of the combination of two lenses is the sum of their separate powers, \(P=P_{1}+P_{2}\)

How large is the image of the Sun on film used in a camera with \((a)\) a 28 -mm-focal-length lens, \((b)\) a 50 -mm-focal-length lens, and (c) a 135 -mm- focal-length lens? (d) If the 50 -mm lens is considered normal for this camera, what relative magnification does cach of the other two lenses provide? The Sun has diameter \(1.4 \times 10^{6} \mathrm{~km},\) and it is \(1.5 \times 10^{8} \mathrm{~km}\) away.

Two converging lenses are placed \(30.0 \mathrm{~cm}\) apart. The focal length of the lens on the right is \(20.0 \mathrm{~cm},\) and the focal length of the lens on the left is \(15.0 \mathrm{~cm} .\) An object is placed to the left of the \(15.0-\mathrm{cm}\) -focal-length lens. A final image from both lenses is inverted and located halfway between the two lenses. How far to the left of the \(15.0-\mathrm{cm}\) -focal length lens is the original object?

When an object is placed \(60.0 \mathrm{~cm}\) from a certain converging lens, it forms a real image. When the object is moved to \(40.0 \mathrm{~cm}\) from the lens, the image moves \(10.0 \mathrm{~cm}\) farther from the lens. Find the focal length of this lens.

A movie star catches a reporter shooting pictures of her at home. She claims the reporter was trespassing. To prove her point, she gives as evidence the film she seized. Her \(1.75-\mathrm{m}\) height is \(8.25 \mathrm{~mm}\) high on the film, and the focal length of the camera lens was \(220 \mathrm{~mm} .\) How far away from the subject was the reporter standing?

As early morning passed toward midday, and the sunlight got more intense, a photographer noted that, if she kept her shutter speed constant, she had to change the \(f\) -number from \(f / 5.6\) to \(f / 16 .\) By what factor had the sunlight intensity increased during that time?

A child has a near point of \(15 \mathrm{~cm}\). What is the maximum magnification the child can obtain using an \(8.5-\mathrm{cm}\) -focallength magnifier? What magnification can a normal eye obtain with the same lens? Which person sees more detail?

A woman can see clearly with her right eye only when objects are between \(45 \mathrm{~cm}\) and \(155 \mathrm{~cm}\) away. Prescription bifocals should have what powers so that she can see distant objects clearly (upper part) and be able to read a book \(25 \mathrm{~cm}\) away (lower part) with her right eye? Assume that the glasses will be \(2.0 \mathrm{~cm}\) from the eye.

What is the magnifying power of a +4.0 -D lens used as a magnifier? Assume a relaxed normal eye.

What is the magnifying power of a \(+4.0-\mathrm{D}\) lens used as a magnifier? Assume a relaxed normal eve.

A physicist lost in the mountains tries to make a telescope using the lenses from his reading glasses. They have powers of \(+2.0 \mathrm{D}\) and \(+4.5 \mathrm{D},\) respectively. \((a)\) What maximum magnification telescope is possible? ( \(b\) ) Which lens should be used as the eyepiece?