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?

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 \(v_{l}\) when (b) \(p=30 \mathrm{~cm}\) (far outside the focal point), (c) \(p=8.0 \mathrm{~cm}\) (just outside the focal point), and (d) \(p=10 \mathrm{~mm}\) (very near the mirror).

Spherical refracting surfaces. An object \(O\) stands on the central axis of a spherical refracting surface. For this situation, each problem in Table \(34-5\) refers to the index of refraction \(n_{1}\) where the object is located, (a) the index of refraction \(n_{2}\) on the other side of the refracting surface, (b) the object distance \(p,(\mathrm{c})\) the radius of curvature \(r\) of the surface, and (d) the image distance \(i\). (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real (R) or virtual (V) and (f) on the same side of the surface as object \(O\) or on the opposite side. \(\begin{array}{lllll}36 & 1.5 & 1.0 & -30 & -7.5\end{array}\)

Spherical refracting surfaces. An object \(O\) stands on the central axis of a spherical refracting surface. For this situation, each problem in Table \(34-5\) refers to the index of refraction \(n_{1}\) where the object is located, (a) the index of refraction \(n_{2}\) on the other side of the refracting surface, (b) the object distance \(p,(\mathrm{c})\) the radius of curvature \(r\) of the surface, and (d) the image distance \(i\). (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real (R) or virtual (V) and (f) on the same side of the surface as object \(O\) or on the opposite side. \(\begin{array}{lllll}7 & 1.5 & 1.0 & +10 & -6.0\end{array}\)

Spherical refracting surfaces. An object \(O\) stands on the central axis of a spherical refracting surface. For this situation, each problem in Table \(34-5\) refers to the index of refraction \(n_{1}\) where the object is located, (a) the index of refraction \(n_{2}\) on the other side of the refracting surface, (b) the object distance \(p,(\mathrm{c})\) the radius of curvature \(r\) of the surface, and (d) the image distance \(i\). (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real (R) or virtual (V) and (f) on the same side of the surface as object \(O\) or on the opposite side. \begin{tabular}{lllll} 38 & \(1.0\) & \(1.5\) & \(+30\) & \(+600\) \\ \hline \end{tabular}

A glass sphere has radius \(R=5.0 \mathrm{~cm}\) and index of refraction 1.6. A paperweight is constructed by slicing through the sphere along a plane that is \(2.0 \mathrm{~cm}\) from the center of the sphere, leaving height \(h=3.0 \mathrm{~cm}\). The paperweight is placed on a table and viewed from directly above by an observer who is distance \(d=8.0 \mathrm{~cm}\) from the tabletop (Fig. 34-39). When viewed through the paperweight, how far away does the tabletop appear to be to the observer?

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 (sin- gle) 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?

Two-lens systems. In Fig. \(34-45\), stick figure \(O\) (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to \(O\), which is at object distance \(p_{1}\). Lens 2 is mounted within the farther boxed region, at distance \(d .\) Each problem in Table \(34-9\) refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by \(\mathrm{C}\) for converging and D for diverging; the number after C or \(\mathrm{D}\) is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance \(i_{2}\) for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification \(M\) for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of lens 2 as object \(O\) or on the opposite side. \(\begin{array}{lllll}\mathbf{8 0} & +10 & \mathrm{C}, 15 & 10 & \mathrm{C}, 8.0\end{array}\)

Two-lens systems. In Fig. \(34-45\), stick figure \(O\) (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to \(O\), which is at object distance \(p_{1}\). Lens 2 is mounted within the farther boxed region, at distance \(d .\) Each problem in Table \(34-9\) refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by \(\mathrm{C}\) for converging and D for diverging; the number after C or \(\mathrm{D}\) is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance \(i_{2}\) for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification \(M\) for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of lens 2 as object \(O\) or on the opposite side. \(\begin{array}{lllll}\mathbf{8 5} & +4.0 & \text { C, } 6.0 & 8.0 & \text { D, } 6.0\end{array}\)

Figure \(34-47 a\) shows the basic structure of a human eye. Light refracts into the eye through the cornea and is then further redirected by a lens whose shape (and thus ability to focus the light) is controlled by muscles. We can treat the cornea and eye lens as a single effective thin lens (Fig. \(34-47 b\) ). A "normal" eye can focus parallel light rays from a distant object \(O\) to a point on the retina at the back of the eye, where processing of the visual information begins. As an object is brought close to the eye, however, the muscles must change the shape of the lens so that rays form an inverted real image on the retina (Fig. \(34-47 c\) ). (a) Suppose that for the parallel rays of Figs. \(34.47 a\) and \(b\), the focal length \(f\) of the effective thin lens of the eye is \(2.50 \mathrm{~cm}\). For an object at distance \(p=\) \(40.0 \mathrm{~cm}\), what focal length \(f^{\prime}\) of the effective lens is required for the object to be seen clearly? (b) Must the eye muscles increase or decrease the radii of curvature of the eye lens to give focal length \(f^{\prime} ?\)

An object is \(10.0 \mathrm{~mm}\) from the objective of a certain compound microscope. The lenses are \(300 \mathrm{~mm}\) apart, and the intermediate image is \(50.0 \mathrm{~mm}\) from the eyepiece. What overall magnification is produced by the instrument?

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?

stick figure \(O\) (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to \(O\), which is at object distance \(p_{1} .\) Lens 2 is mounted within the middle boxed region, at distance \(d_{12}\) from lens \(1 .\) Lens 3 is mounted in the farthest boxed region, at distance \(d_{23}\) from lens \(2 .\) Each problem in Table \(34-10\) refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by \(C\) for converging and \(D\) for diverging: the number after \(\mathrm{C}\) or \(\mathrm{D}\) is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance \(i_{2}\) for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification \(M\) for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of lens 3 as object \(O\) or on the opposite side. \(\begin{array}{lllllll}95 & +12 & \mathrm{C}, 8.0 & 28 & \mathrm{C}, 6.0 & 8.0 & \mathrm{C}, 6.0\end{array}\)

stick figure \(O\) (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to \(O\), which is at object distance \(p_{1} .\) Lens 2 is mounted within the middle boxed region, at distance \(d_{12}\) from lens \(1 .\) Lens 3 is mounted in the farthest boxed region, at distance \(d_{23}\) from lens \(2 .\) Each problem in Table \(34-10\) refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by \(C\) for converging and \(D\) for diverging: the number after \(\mathrm{C}\) or \(\mathrm{D}\) is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance \(i_{2}\) for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification \(M\) for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of lens 3 as object \(O\) or on the opposite side. \(\begin{array}{lllllll}97 & +18 & \mathrm{C}, 6.0 & 15 & \mathrm{C}, 3.0 & 11 & \mathrm{C}, 3.0\end{array}\)

stick figure \(O\) (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to \(O\), which is at object distance \(p_{1} .\) Lens 2 is mounted within the middle boxed region, at distance \(d_{12}\) from lens \(1 .\) Lens 3 is mounted in the farthest boxed region, at distance \(d_{23}\) from lens \(2 .\) Each problem in Table \(34-10\) refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by \(C\) for converging and \(D\) for diverging: the number after \(\mathrm{C}\) or \(\mathrm{D}\) is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance \(i_{2}\) for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification \(M\) for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of lens 3 as object \(O\) or on the opposite side. \begin{tabular}{lllllll} \(\mathbf{1 0 0}\) & \(+4.0\) & \(\mathbf{C}, 6.0\) & \(8.0\) & \(\mathrm{D}, 4.0\) & \(5.7\) & \(\mathrm{D}, 12\) \\ \hline \end{tabular}

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 \(H_{l}=2.0 \mathrm{H}\). 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 \(H_{I}=0.50 \mathrm{H}\). 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.)

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 small cup of green tea is positioned on the central axis of a spherical mirror. The lateral magnification of the cup is \(+0.250\), and the distance between the mirror and its focal point is \(2.00 \mathrm{~cm}\). (a) What is the distance between the mirror and the image it produces? (b) Is the focal length positive or negative? (c) Is the image real or virtual?

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?

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.)