33. Geometric Optics

How many uncoated thin lenses in an optical instrument would reduce the amount of light passing through the instrument to 50% or less? (Assume the same transmission percent at each of the two surfaces—see page 1008.)

A bright object and a viewing screen are separated by a distance of 86.0 cm. At what location(s) between the object and the screen should a lens of focal length 16.0 cm be placed in order to produce a sharp image on the screen? [Hint: First draw a diagram.]

An object is placed so it is touching one surface of a thin lens. Where is the image in this case (i.e., when dₒ approaches 0)?

Students in a physics lab are assigned to find the location where a bright object may be placed in order that a converging lens with ƒ = 12 cm will produce an image three times the size of the object. Two students complete the assignment at different times using identical equipment, but when they compare notes later, they discover that their answers for the object distance are not the same. Explain why they do not necessarily need to repeat the lab, and justify your response with a calculation.

(III) A bright object is placed on one side of a converging lens of focal length f, and a white screen for viewing the image is on the opposite side. The distance d_T = d_i + d_o between the object and the screen is kept fixed, but the lens can be moved. Determine a formula for the distance between the two lens positions in part (a), and the ratio of the image sizes.

(II) A diverging lens is placed next to a converging lens of focal length ƒ_C , as in Fig. 33–14. If ƒ_T represents the focal length of the combination, show that the focal length of the diverging lens, ƒ_D , is given by

1/ƒ_D = (1/ƒ_T) - (1/ƒ_C)

(II) In a film projector, the film acts as the object whose image is projected on a screen (Fig. 33–46). If a 105-mm-focal-length lens is to project an image on a screen 22.5 m away, how far from the lens should the film be? If the film is 24 mm wide, how wide will the picture be on the screen?

Figure 33–51 was taken from the NIST Laboratory (National Institute of Standards and Technology) in Boulder, CO, 2.0 km from the hiker in the photo. The Sun’s image was 15 mm across on the film. Estimate the focal length of the camera lens (actually a telescope). The Sun has diameter 1.4 x 10⁶ km, and it is 1.5 x 10⁸ km away.

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(II) A planoconvex lens (Fig. 33–2a) has one flat surface and the other has R = 15.3 cm. This lens is used to view a red and yellow object which is 62.0 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?

The focal length f of a converging lens can be found by placing an object of known size at various locations in front of the lens and measuring the resulting real-image distances dᵢ and their associated magnifications m (minus sign indicates that image is inverted). The data taken in such an experiment are given here:

(a) Show algebraically that a graph of m vs. dᵢ should produce a straight line. What are the theoretically expected values for the slope and the y-intercept of this line? [Hint: dₒ is not constant.] (b) Using the data above, graph m vs. dᵢ 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?

BIO The Lens of the Eye. The crystalline lens of the human eye is a double-convex lens made of material having an index of refraction of 1.44 (although this varies). Its focal length in air is about 8.0 mm, which also varies. We shall assume that the radii of curvature of its two surfaces have the same magnitude. Find the radii of curvature of this lens.

A 1.0-cm-tall object is 110 cm from a screen. A diverging lens with focal length -20 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 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. What are the height and orientation of the final image?

What is the focal length of a second lens that could be placed in contact with the first lens to provide an overall power of 30 D?

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 of the objective to a real on the sensor of a digital camera. Suppose the sensor is 22.5 mm wide, a typical value, with 4.0 μm x 4.0 μm pixels. The photo of a cell is 120 pixels in diameter. What is the cell’s actual diameter, in μm?