Chapter 34

You look through a camera toward an image of a hummingbird in a plane mirror. The camera is \(4.30 \mathrm{~m}\) in front of the mirror. The bird is at camera level, \(5.00 \mathrm{~m}\) to your right and \(3.30 \mathrm{~m}\) from the mirror. What is the distance between the camera and the apparent position of the bird's image in the mirror?

A moth at about eye level is \(10 \mathrm{~cm}\) in front of a plane mirror; you are behind the moth, \(30 \mathrm{~cm}\) from the mirror. What is the distance between your eyes and the apparent position of the moth's image in the mirror?

A concave shaving mirror has a radius of curvature of \(35.0 \mathrm{~cm}\). It is positioned so that the (upright) image of a man's face is 2.50 times the size of the face. How far is the mirror from the face?

Object \(O\) stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table \(34-4\) refers to (a) the type of mirror, (b) the focal distance \(f,\) (c) the radius of curvature \(r\), (d) the object distance \(p,\) (e) the image distance \(i\), and (f) the lateral magnification \(m\). (All distances are in centimeters.) It also refers to whether \((\mathrm{g})\) the image is real \((\mathrm{R})\) or virtual (V), (h) inverted (I) or noninverted (NI) from \(O\), and (i) on the same side of the mirror as object \(O\) or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.

(a) A luminous point is moving at speed \(v_{O}\) toward a spherical mirror with radius of curvature \(r\), along the central axis of the mirror. Show that the image of this point is moving at speed \(v_{I}=-\left(\frac{r}{2 p-r}\right)^{2} v_{O}\) where \(p\) is the distance of the luminous point from the mirror at any given time. Now assume the mirror is concave, with \(r=15 \mathrm{~cm}\), and let \(v_{O}=5.0 \mathrm{~cm} / \mathrm{s} .\) Find \(v_{I}\) when \((\mathrm{b}) p=30 \mathrm{~cm}\) (far outside the focal point \(),(\mathrm{c}) p=8.0 \mathrm{~cm}\) (just outside the focal point), and (d) \(p=10 \mathrm{~mm}\) (very near the mirror).

A lens is made of glass having an index of refraction of 1.5. One side of the lens is flat, and the other is convex with a radius of curvature of \(20 \mathrm{~cm} .\) (a) Find the focal length of the lens. (b) If an object is placed \(40 \mathrm{~cm}\) in front of the lens, where is the image?

A movie camera with a (single) lens of focal length \(75 \mathrm{~mm}\) takes a picture of a person standing \(27 \mathrm{~m}\) away. If the person is \(180 \mathrm{~cm}\) tall, what is the height of the image on the film?

A double-convex lens is to be made of glass with an index of refraction of \(1.5 .\) One surface is to have twice the radius of curvature of the other and the focal length is to be \(60 \mathrm{~mm}\). What is the (a) smaller and (b) larger radius?

An illuminated slide is held \(44 \mathrm{~cm}\) from a screen. How far from the slide must a lens of focal length \(11 \mathrm{~cm}\) be placed (between the slide and the screen) to form an image of the slide's picture on the screen?

Object \(O\) stands on the central axis of a thin symmetric lens. For this situation, each problem in Table \(34-6\) gives object distance \(p\) (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance \(i\) and (b) the lateral magnification \(m\) of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of the lens as object \(O\) or on the opposite side.

If the angular magnification of an astronomical telescope is 36 and the diameter of the objective is \(75 \mathrm{~mm},\) what is the minimum diameter of the eyepiece required to collect all the light entering the objective from a distant point source on the telescope axis?

Someone with a near point \(P_{n}\) of \(25 \mathrm{~cm}\) views a thimble through a simple magnifying lens of focal length \(10 \mathrm{~cm}\) by placing the lens near his eye. What is the angular magnification of the thimble if it is positioned so that its image appears at (a) \(P_{n}\) and (b) infinity?

The formula \(1 / p+1 / i=1 / f\) is called the Gaussian form of the thin-lens formula. Another form of this formula, the Newtonian form, is obtained by considering the distance \(x\) from the object to the first focal point and the distance \(x^{\prime}\) from the second focal point to the image. Show that \(x x^{\prime}=f^{2}\) is the Newtonian form of the thin-lens formula.

Two thin lenses of focal lengths \(f_{1}\) and \(f_{2}\) are in contact and share the same central axis. Show that, in image formation, they are equivalent to a single thin lens for which the focal length is \(f=f_{1} f_{2} /\left(f_{1}+f_{2}\right)\)

Two plane mirrors are placed parallel to each other and \(40 \mathrm{~cm}\) apart. An object is placed \(10 \mathrm{~cm}\) from one mirror. Determine the (a) smallest, (b) second smallest, (c) third smallest (occurs twice), and (d) fourth smallest distance between the object and image of the object.

A fruit fly of height \(H\) sits in front of lens 1 on the central axis through the lens. The lens forms an image of the fly at a distance \(d=20 \mathrm{~cm}\) from the fly; the image has the fly's orientation and height \(H_{I}=2.0 \mathrm{H}\). What are (a) the focal length \(f_{1}\) of the lens and (b) the object distance \(p_{1}\) of the fly? The fly then leaves lens 1 and sits in front of lens \(2,\) which also forms an image at \(d=20 \mathrm{~cm}\) that has the same orientation as the fly, but now \(H_{I}=0.50 H\). What are (c) \(f_{2}\) and (d) \(p_{2} ?\)

Figure 34-56 shows a beam expander made with two coaxial converging lenses of focal lengths \(f_{1}\) and \(f_{2}\) and separation \(d=f_{1}+f_{2}\) The device can expand a laser beam while keeping the light rays in the beam parallel to the central axis through the lenses Suppose a uniform laser beam of width \(W_{i}=2.5 \mathrm{~mm}\) and intensity \(I_{i}=9.0 \mathrm{~kW} / \mathrm{m}^{2}\) enters a beam expander for which \(f_{1}=12.5 \mathrm{~cm}\) and \(f_{2}=30.0 \mathrm{~cm} .\) What are (a) \(W_{f}\) and (b) \(I_{f}\) of the beam leaving the expander? (c) What value of \(d\) is needed for the beam expander if lens 1 is replaced with a diverging lens of focal length \(f_{1}=-26.0 \mathrm{~cm} ?\)

A pinhole camera has the hole a distance \(12 \mathrm{~cm}\) from the film plane, which is a rectangle of height \(8.0 \mathrm{~cm}\) and width \(6.0 \mathrm{~cm} .\) How far from a painting of dimensions \(50 \mathrm{~cm}\) by \(50 \mathrm{~cm}\) should the camera be placed so as to get the largest complete image possible on the film plane?

Light travels from point \(A\) to point \(B\) via reflection at point \(O\) on the surface of a mirror. Without using calculus, show that length \(A O B\) is a minimum when the angle of incidence \(\theta\) is equal to the angle of reflection \(\phi .\) (Hint: Consider the image of \(A\) in the mirror.

A point object is \(10 \mathrm{~cm}\) away from a plane mirror, and the eye of an observer (with pupil diameter \(5.0 \mathrm{~mm}\) ) is \(20 \mathrm{~cm}\) away. Assuming the eye and the object to be on the same line perpendicular to the mirror surface, find the area of the mirror used in observing the reflection of the point. (Hint: Adapt Fig. \(34-4 .)\)

Show that the distance between an object and its real image formed by a thin converging lens is always greater than or equal to four times the focal length of the lens.

A luminous object and a screen are a fixed distance \(D\) apart. (a) Show that a converging lens of focal length \(f,\) placed between object and screen, will form a real image on the screen for two lens positions that are separated by a distance \(d=\sqrt{D(D-4 f)}\) (b) Show that \(\left(\frac{D-d}{D+d}\right)^{2}\) gives the ratio of the two image sizes for these two positions of the lens.

An eraser of height \(1.0 \mathrm{~cm}\) is placed \(10.0 \mathrm{~cm}\) in front of a two-lens system. Lens 1 (nearer the eraser) has focal length \(f_{1}=\) \(-15 \mathrm{~cm},\) lens 2 has \(f_{2}=12 \mathrm{~cm},\) and the lens separation is \(d=12 \mathrm{~cm}\) For the image produced by lens \(2,\) what are (a) the image distance \(i_{2}\) (including sign), (b) the image height, (c) the image type (real or virtual), and (d) the image orientation (inverted relative to the eraser or not inverted)?

A peanut is placed \(40 \mathrm{~cm}\) in front of a two-lens system: lens 1 (nearer the peanut) has focal length \(f_{1}=+20 \mathrm{~cm},\) lens 2 has \(f_{2}=-15 \mathrm{~cm},\) and the lens separation is \(d=10 \mathrm{~cm} .\) For the image produced by lens \(2,\) what are (a) the image distance \(i_{2}\) (including sign), (b) the image orientation (inverted relative to the peanut or not inverted), and (c) the image type (real or virtual)? (d) What is the net lateral magnification?

A coin is placed \(20 \mathrm{~cm}\) in front of a two-lens system. Lens 1 (nearer the coin) has focal length \(f_{1}=+10 \mathrm{~cm},\) lens 2 has \(f_{2}=\) \(+12.5 \mathrm{~cm},\) and the lens separation is \(d=30 \mathrm{~cm} .\) For the image produced by lens \(2,\) what are (a) the image distance \(i_{2}\) (including sign), (b) the overall lateral magnification, (c) the image type (real or virtual), and (d) the image orientation (inverted relative to the coin or not inverted)?

An object is \(20 \mathrm{~cm}\) to the left of a thin diverging lens that has a \(30 \mathrm{~cm}\) focal length. (a) What is the image distance \(i ?\) (b) Draw a ray diagram showing the image position.

One end of a long glass rod \((n=1.5)\) is a convex surface of radius \(6.0 \mathrm{~cm} .\) An object is located in air along the axis of the rod, at a distance of \(10 \mathrm{~cm}\) from the convex end. (a) How far apart are the object and the image formed by the glass rod? (b) Within what range of distances from the end of the rod must the object be located in order to produce a virtual image?

A short straight object of length \(L\) lies along the central axis of a spherical mirror, a distance \(p\) from the mirror. (a) Show that its image in the mirror has a length \(L^{\prime},\) where \(L^{\prime}=L\left(\frac{f}{p-f}\right)^{2}\) (Hint: Locate the two ends of the object.) (b) Show that the longitudinal magnification \(m^{\prime}\left(=L^{\prime} / L\right)\) is equal to \(m^{2},\) where \(m\) is the lateral magnification.

Prove that if a plane mirror is rotated through an angle \(\alpha\), the reflected beam is rotated through an angle \(2 \alpha .\) Show that this result is reasonable for \(\alpha=45^{\circ}\).

An object is \(30.0 \mathrm{~cm}\) from a spherical mirror, along the mirror's central axis. The mirror produces an inverted image with a lateral magnification of absolute value \(0.500 .\) What is the focal length of the mirror?

A concave mirror has a radius of curvature of \(24 \mathrm{~cm} .\) How far is an object from the mirror if the image formed is (a) virtual and 3.0 times the size of the object, (b) real and 3.0 times the size of the object, and (c) real and \(1 / 3\) the size of the object?

A millipede sits \(1.0 \mathrm{~m}\) in front of the nearest part of the surface of a shiny sphere of diameter \(0.70 \mathrm{~m}\). (a) How far from the surface does the millipede's image appear? (b) If the millipede's height is \(2.0 \mathrm{~mm},\) what is the image height? (c) Is the image inverted?

A corner reflector, much used in optical, microwave, and other applications, consists of three plane mirrors fastened together to form the corner of a cube. Show that after three reflections, an incident ray is returned with its direction exactly reversed.

A cheese enchilada is \(4.00 \mathrm{~cm}\) in front of a converging lens. The magnification of the enchilada is \(-2.00 .\) What is the focal length of the lens?

A grasshopper hops to a point on the central axis of a spherical mirror. The absolute magnitude of the mirror's focal length is \(40.0 \mathrm{~cm},\) and the lateral magnification of the image produced by the mirror is \(+0.200 .\) (a) Is the mirror convex or concave? (b) How far from the mirror is the grasshopper?

Suppose the farthest distance a person can see without visual aid is \(50 \mathrm{~cm} .\) (a) What is the focal length of the corrective lens that will allow the person to see very far away? (b) Is the lens converging or diverging? (c) The power \(P\) of a lens (in diopters) is equal to \(1 / f,\) where \(f\) is in meters. What is \(P\) for the lens?

A simple magnifier of focal length \(f\) is placed near the eye of someone whose near point \(P_{n}\) is \(25 \mathrm{~cm} .\) An object is positioned so that its image in the magnifier appears at \(P_{n}\). (a) What is the angular magnification of the magnifier? (b) What is the angular magnification if the object is moved so that its image appears at infinity? For \(f=10 \mathrm{~cm},\) evaluate the angular magnifications of (c) the situation in (a) and (d) the situation in (b). (Viewing an image at \(P_{n}\) requires effort by muscles in the eye, whereas viewing an image at infinity requires no such effort for many people.)