Chapter 33

A 50-year-old man uses \(+2.5-\mathrm{D}\) lenses to read a newspaper \(25 \mathrm{~cm}\) away. Ten years later, he must hold the paper \(32 \mathrm{~cm}\) away to see clearly with the same lenses. What power lenses does he need now in order to hold the paper \(25 \mathrm{~cm}\) away? (Distances are measured from the lens.)

An object is moving toward a converging lens of focal length \(f\) with constant speed \(v_{\mathrm{o}}\) such that its distance \(d_{\mathrm{o}}\) from the lens is always greater than \(f\). (a) Determine the velocity \(v_{i}\) of the image as a function of \(d_{0} .(b)\) Which direction (toward or away from the lens) does the image move? (c) For what \(d_{\mathrm{o}}\) does the image's speed equal the object's speed?

The objective lens and the eyepiece of a telescope are spaced \(85 \mathrm{~cm}\) apart. If the eyepiece is \(+23 \mathrm{D},\) what is the total magnification of the telescope?

Two converging lenses, one with \(f=4.0 \mathrm{cm}\) and the other with \(f=44 \mathrm{cm},\) are made into a telescope. \((a)\) What are the length and magnification? Which lens should be the eyepiece? \((b)\) Assume these lenses are now combined to make a microscope; if the magnification needs to be \(25 \times\) , how long would the microscope be?

Sam purchases +3.50 -D eyeglasses which correct his faulty vision to put his near point at \(25 \mathrm{~cm}\). (Assume he wears the lenses \(2.0 \mathrm{~cm}\) from his eyes.) (a) Calculate the focal length of Sam's glasses. (b) Calculate Sam's near point without glasses. (c) Pam, who has normal eyes with near point at \(25 \mathrm{~cm},\) puts on Sam's glasses. Calculate Pam's near point with Sam's glasses on.

Sam purchases \(+3.50-\mathrm{D}\) eyeglasses which correct his faulty vision to put his near point at 25 \(\mathrm{cm}\) . (Assume he wears the lenses 2.0 \(\mathrm{cm}\) from his eyes.) (a) Calculate the focal length of Sam's glasses. (b) Calculate Sam's near point without glasses. (c) Pam, who has normal eyes with near point at \(25 \mathrm{cm},\) puts on Sam's glasses. Calculate Pam's near point with Sam's glasses on.

The proper functioning of certain optical devices (e.g., optical fibers and spectrometers) requires that the input light be a collection of diverging rays within a cone of halfangle \(\theta\) (Fig. \(33-50\) ). If the light initially exists as a collimated beam (i.e., parallel rays), show that a single lens of focal length \(f\) and diameter \(D\) can be used to create the required input light if \(D / f=2 \tan \theta .\) If \(\theta=3.5^{\circ}\) for a certain spectrometer, what focal length lens should be used if the lens diameter is \(5.0 \mathrm{~cm} ?\)

In a science-fiction novel, an intelligent ocean-dwelling creature's eye functions underwater with a near point of 25 \(\mathrm{cm} .\) This creature would like to create an underwater magnifier out of a thin plastic container filled with air. What shape should the air-filled plastic container have (i.e., determine radii of curvature of its surfaces) in order for it to be used by the creature as a \(3.0 \times\) magnifier? Assume the eye is focused at its near point.

A telephoto lens system obtains a large magnification in a compact package. A simple such system can be constructed out of two lenses, one converging and one diverging, of focal lengths \(f_{1}\) and \(f_{2}=-\frac{1}{2} f_{1},\) respectively, separated by a distance \(\ell=\frac{3}{4} f_{1}\) as shown in Fig. \(51 .\) (a) For a distant object located at distance \(d_{\mathrm{o}}\) from the first lens, show that the first lens forms an image with magnification \(m_{1} \approx-f_{1} / d_{0}\) located very close to its focal point. Go on to show that the total magnification for the two-lens system is \(m \approx-2 f_{1} / d_{0} .(b)\) For an object located at infinity, show that the two-lens system forms an image that is a distance \(\frac{5}{4} f_{1}\) behind the first lens. (c) A single 250 -mm-focal-length lens would have to be mounted about 250 \(\mathrm{mm}\) from a camera's film in order to produce an image of a distant object at \(d_{\mathrm{o}}\) with magnification \(-(250 \mathrm{mm}) / d_{\mathrm{o}} .\) To produce an image of this object with the same magnification using the two-lens system, what value of \(f_{1}\) should be used and how far in front of the film should the first lens be placed? How much smaller is the "focusing length" (i.e., first lens-to-final image distance) of this two-lens system in comparison with the \(250-\mathrm{mm}\) "focusing length" of the equivalent single lens?

(III) In the "magnification" method, the focal length \(f\) of a converging lens is found by placing an object of known size at various locations in front of the lens and measuring the resulting real-image distances \(d_{i}\) and their associated magnifications \(m\) (minus sign indicates that image is inverted). The data taken in such an experiment are given here:\(\begin{array}{rrrrrr}{d_{\mathrm{i}}(\mathrm{cm})} & {20} & {25} & {30} & {35} & {40} \\ {m} & {-0.43} & {-0.79} & {-1.14} & {-1.50} & {-1.89}\end{array}\) (a) Show analytically that a graph of \(m\) vs. \(d_{\text { i should }}\) produce a straight line. What are the theoretically expected values for the slope and the \(y\) -intercept of this line? [Hint: \(d_{\mathrm{o}}\) is not constant.] \((b)\) Using the data above, graph \(m\) vs. \(d_{\mathrm{i}}\) and show that a straight line does indeed result. Use the slope of this line to determine the focal length of the lens. Does the \(y\) -intercept of your plot have the expected value? (c) In performing such an experiment, one has the practical problem of locating the exact center of the lens since \(d_{\mathrm{i}}\) must be measured from this point. Imagine, instead, that one measures the image distance \(d\) from the back surface of the lens, which is a distance \(\ell\) from the lens's center. Then, \(d_{i}=d_{1}^{\prime}+\ell .\) Show that, when implementing the magnification method in this fashion, a plot of \(m\) vs.di will still result in a straight line. How can \(f\) be determined from this straight line?