|| The rays leaving the two-component optical system of FIGURE P35.26 produce two distinct images of the 1.0-cm-tall object. What are the position (relative to the lens), orientation, and height of each image?

FIGURE P35.26

||| In FIGURE P35.27, parallel rays from the left are focused to a point at two locations on the optical axis. Find the position of each location, giving your answer as a distance left or right of the lens.

FIGURE P35.27

| A common optical instrument in a laser laboratory is a beam expander. One type of beam expander is shown in FIGURE P35.28. The parallel rays of a laser beam of width $w_1$ enter from the left.

1. For what lens spacing $d$ does a parallel laser beam exit from the right?

2. What is the width $w_2$ of the exiting laser beam?

FIGURE P35.28

| A common optical instrument in a laser laboratory is a beam expander. One type of beam expander is shown in FIGURE P35.29. The parallel rays of a laser beam of width $w_1$ enter from the left.

1. For what lens spacing $d$ does a parallel laser beam exit from the right?

2. What is the width $w_2$ of the exiting laser beam?

FIGURE P35.29

||| In FIGURE P35.30, what are the position, height, and orientation of the final image? Give the position as a distance to the right or left of the lens.

FIGURE P35.30

|| A 1.0-cm-tall object is 110 cm from a screen. A diverging lens with focal length $-20\text{cm}$ is 20 cm in front of the object. What are the focal length and distance from the screen of a second lens that will produce a well-focused, 2.0-cm-tall image on the screen?

|| A 15-cm-focal-length converging lens is 20 cm to the right of a 7.0-cm-focal-length converging lens. A 1.0-cm-tall object is distance $L$ to the left of the 7.0-cm-focal-length lens.

1. For what value of $L$ is the final image of this two-lens system halfway between the two lenses?

2. What are the height and orientation of the final image?

|| BIO Yang can focus on objects 150 cm away with a relaxed eye. With full accommodation, she can focus on objects 20 cm away. After her eyesight is corrected for distance vision, what will her near point be while wearing her glasses?

|| BIO The cornea, a boundary between the air and the aqueous humor, has a 3.0 cm focal length when acting alone. What is its radius of curvature?

|| Mars (6800 km diameter) is viewed through a telescope on a night when it is $1.1\times {10}^8\text{km}$ from the earth. Its angular size as seen through the eyepiece is $0.50\text{°}$, the same size as the full moon seen by the naked eye. If the eyepiece focal length is 25 mm, how long is the telescope?

| You’ve been asked to build a telescope from a $2.0\times$ magnifying lens and a $5.0\times$ magnifying lens.

1. What is the maximum magnification you can achieve?

2. Which lens should be used as the objective? Explain.

3. What will be the length of your telescope?

| Marooned on a desert island and with a lot of time on your hands, you decide to disassemble your glasses to make a crude telescope with which you can scan the horizon for rescuers. Luckily you’re farsighted, and, like most people, your two eyes have different lens prescriptions. Your left eye uses a lens of power $+4.5\text{D,}$ and your right eye’s lens is $+3.0\text{D}$.

1. Which lens should you use for the objective and which for the eyepiece? Explain.

2. What will be the magnification of your telescope?

3. How far apart should the two lenses be when you focus on distant objects?

|| A microscope with a tube length of 180 mm achieves a total magnification of $800\times$ with a $40\times$ objective and a $20\times$ eyepiece. The microscope is focused for viewing with a relaxed eye. Approximately how far is the sample from the objective lens?

|| BIO Modern microscopes are more likely to use a camera than human viewing. This is accomplished by replacing the eyepiece in Figure 35.14 with a photo-ocular that focuses the image of the objective to a real image on the sensor of a digital camera. Suppose the sensor is 22.5 mm wide, a typical value, with $4.0\mu \text{m}\times 4.0\mu \text{m}$ pixels.

1. What is the field of view? That is, what width on the microscope stage, in mm, fills the sensor?

2. The photo of a cell is 120 pixels in diameter. What is the cell’s actual diameter, in $\mu \text{m}?$

|| A simple and relatively inexpensive microscope eyepiece is the Ramsden eyepiece shown in FIGURE P35.40. Two plano-convex lenses have their curved surfaces facing each other, which a more advanced analysis shows is the orientation that minimizes spherical aberration. That same analysis finds that chromatic aberration is minimized with lens spacing $L=\frac{1}{2}(f_1+f_2)$. Your task is to design a $10\times$ Ramsden eyepiece in which the first lens has a focal length of 30 mm. What are (a) the focal length and (b) the spacing of the second lens?

FIGURE P35.40

| White light is incident onto a $30\text{°}$ prism at the $40\text{°}$ angle shown in FIGURE P35.41. Violet light emerges perpendicular to the rear face of the prism. The index of refraction of violet light in this glass is 2.0% larger than the index of refraction of red light. At what angle $\varphi$ does red light emerge from the rear face?

FIGURE P35.41

| High-power lasers are used to cut and weld materials by focusing the laser beam to a very small spot. This is like using a magnifying lens to focus the sun’s light to a small spot that can burn things. As an engineer, you have designed a laser cutting device in which the material to be cut is placed 5.0 cm behind the lens. You have selected a high-power laser with a wavelength of $1.06\mu \text{m}$. Your calculations indicate that the laser must be focused to a $5.0\text{-}\mu \text{m}\text{-diameter}$ spot in order to have sufficient power to make the cut. What is the minimum diameter of the lens you must install?

||| BIO Once dark adapted, the pupil of your eye is approximately 7 mm in diameter. The headlights of an oncoming car are 120 cm apart. If the lens of your eye is diffraction limited, at what distance are the two headlights marginally resolved? Assume a wavelength of 600 nm and that the index of refraction inside the eye is 1.33. (Your eye is not really good enough to resolve headlights at this distance, due both to aberrations in the lens and to the size of the receptors in your retina, but it comes reasonably close.)

|| INT The resolution of a digital camera is limited by two factors: diffraction by the lens, a limit of any optical system, and the fact that the sensor is divided into discrete pixels. Consider a typical point-and-shoot camera that has a 20-mm-focal-length lens and a sensor with $2.5\mu \text{m}\times 2.5\mu \text{m}$ pixels.

1. First, assume an ideal, diffractionless lens. At a distance of 100 m, what is the smallest distance, in cm, between two point sources of light that the camera can barely resolve? In answering this question, consider what has to happen on the sensor to show two image points rather than one. You can use $s\prime =f$ because $s\gg f$.

2. You can achieve the pixel-limited resolution of part a only if the diffraction width of each image point is no greater than 1 pixel in diameter. For what lens diameter is the minimum spot size equal to the width of a pixel? Use 600 nm for the wavelength of light.

3. What is the f-number of the lens for the diameter you found in part b? Your answer is a quite realistic value of the f-number at which a camera transitions from being pixel limited to being diffraction limited. For $f$-numbers smaller than this (larger-diameter apertures), the resolution is limited by the pixel size and does not change as you change the aperture. For f-numbers larger than this (smaller-diameter apertures), the resolution is limited by diffraction, and it gets worse as you “stop down” to smaller apertures.

|| The Hubble Space Telescope has a mirror diameter of 2.4 m. Suppose the telescope is used to photograph stars near the center of our galaxy, 30,000 light years away, using red light with a wavelength of 650 nm.

1. What’s the distance (in km) between two stars that are marginally resolved? The resolution of a reflecting telescope is calculated exactly the same as for a refracting telescope.

2. For comparison, what is this distance as a multiple of the distance of Jupiter from the sun?

|| Alpha Centauri, the nearest star to our solar system, is 4.3 light years away. Assume that Alpha Centauri has a planet with an advanced civilization. Professor Dhg, at the planet’s Astronomical Institute, wants to build a telescope with which he can find out whether any planets are orbiting our sun.

1. What is the minimum diameter for an objective lens that will just barely resolve Jupiter and the sun? The radius of Jupiter’s orbit is 780 million km. Assume $\text{λ}=600\text{nm}$.

2. Building a telescope of the necessary size does not appear to be a major problem. What practical difficulties might prevent Professor Dhg’s experiment from succeeding?

||| CALC A beam of white light enters a transparent material. Wavelengths for which the index of refraction is $n$ are refracted at angle ${\theta }_2$. Wavelengths for which the index of refraction is $n+\delta n$, where $\delta n\ll n$, are refracted at angle ${\theta }_2+\delta \theta$.

1. Show that the angular separation of the two wavelengths, in radians, is $\delta \theta =-\left(\delta n/n\right)\text{tan }{\theta }_2$.

2. A beam of white light is incident on a piece of glass at $30.0\text{°}$. Deep violet light is refracted $0.28\text{°}$ more than deep red light. The index of refraction for deep red light is known to be 1.552. What is the index of refraction for deep violet light?

||| Your task in physics laboratory is to make a microscope from two lenses. One lens has a focal length of 2.0 cm, the other 1.0 cm. You plan to use the more powerful lens as the objective, and you want the eyepiece to be 16 cm from the objective.

1. For viewing with a relaxed eye, how far should the sample be from the objective lens?

2. What is the magnification of your microscope?

||| The lens shown in FIGURE CP35.49 is called an achromatic doublet, meaning that it has no chromatic aberration. The left side is flat, and all other surfaces have radii of curvature $R$.

1. For parallel light rays coming from the left, show that the effective focal length of this two-lens system is $f=R/(2n_2-n_1-1)$, where $n_1$ and $n_2$ are, respectively, the indices of refraction of the diverging and the converging lenses. Don’t forget to make the thin-lens approximation.

2. Because of dispersion, either lens alone would focus red rays and blue rays at different points. Define $\text{Δ}{n}_1$ and $\text{Δ}{n}_2$ as $n_{\text{blue}}-n_{\text{red}}$ for the two lenses. What value of the ratio $\text{Δ}{n}_1\text{/}\text{Δ}{n}_2$ makes $f_{\text{blue}}=f_{\text{red}}$ for the two-lens system? That is, the two-lens system does not exhibit chromatic aberration.

3. Indices of refraction for two types of glass are given in the table. To make an achromatic doublet, which glass should you use for the converging lens and which for the diverging lens? Explain.

   	$n{}_{\text{blue}}$	$n{}_{\text{red}}$
   Crown glass	1.525	1.517
   Flint glass	1.632	1.616

4. What value of $R$ gives a focal length of 10.0 cm?

   FIGURE CP35.49

   

   FIGURE CP35.49 Full Alternative Text

||| FIGURE CP35.50 shows a lens combination in which the lens separation $L$ is less than the focal length of the converging lens. The procedure for combination lenses is to let the image of the first lens be the object for the second lens, but in this case the image of the first lens—shown as a dot—is on the far side of the second lens. This is called a virtual object, a point that light rays are converging toward but never reach. The top half of Figure CP35.50 shows that the converging rays are refracted again by the diverging lens and come to a focus farther to the right. The procedure for combination lenses will continue to work if we use a negative object distance for a virtual object.

1. For input light rays parallel to the axis from a very distant object, find the point at which the rays are focused for lens separations of 5 cm and 10 cm. Give your answers as distances to the right of the diverging lens.

2. Equation 35.1 defined the effective focal length $f_{\text{eff}}$ of a lens combination, but we didn’t discuss how it is used. Although an actual ray refracts twice, once at each lens, we can extend the output rays leftward to where they need to bend only once in a plane called the principal plane. The principal plane is similar to the lens plane of a single lens, where a single bend occurs, but the principal plane generally does not coincide with the physical lens; it’s just a mathematical plane in space. The effective focal length is measured from the principal plane, so parallel input rays are focused at distance $f_{\text{eff}}$ beyond the principal plane. Find the positions of the principal planes for lens separations of 5 cm and 10 cm. Give your answers as distances to the left of the diverging lens.

FIGURE CP35.50