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Ch. 33 - Lenses and Optical Instruments
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 33 - Lenses and Optical InstrumentsProblem 24
Chapter 32, Problem 24

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


1/ƒD = (1/ƒT) - (1/ƒC)

Verified step by step guidance
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Step 1: Understand the problem setup. The system consists of two lenses: a converging lens with focal length ƒ_C and a diverging lens with focal length ƒ_D. The combination of these lenses results in an effective focal length ƒ_T. The goal is to derive the relationship between ƒ_D, ƒ_C, and ƒ_T.
Step 2: Recall the formula for the effective focal length of a combination of two lenses placed in contact. The formula is given by: 1ƒ=1ƒC+1ƒD, where ƒ is the effective focal length of the system.
Step 3: Substitute the effective focal length ƒ_T for ƒ in the formula. This gives: 1ƒT=1ƒC+1ƒD.
Step 4: Rearrange the equation to isolate the term involving ƒ_D. Subtract 1ƒC from both sides: 1ƒD=1ƒT-1ƒC.
Step 5: Interpret the result. The derived equation shows that the reciprocal of the focal length of the diverging lens, ƒ_D, is equal to the difference between the reciprocals of the effective focal length of the combination, ƒ_T, and the focal length of the converging lens, ƒ_C. This relationship can be used to calculate ƒ_D if ƒ_T and ƒ_C are known.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Lens Formula

The lens formula relates the focal length (f), object distance (u), and image distance (v) for lenses. It is expressed as 1/f = 1/v - 1/u. This formula is fundamental in optics, allowing us to understand how lenses form images based on their focal lengths and the positions of objects.
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Lens Maker Equation

Focal Length of Lenses

The focal length of a lens is the distance from the lens to the focal point, where parallel rays of light converge (for converging lenses) or appear to diverge (for diverging lenses). The sign of the focal length indicates the type of lens: positive for converging lenses and negative for diverging lenses, which is crucial for calculations involving multiple lenses.
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Ray Diagrams for Converging Lenses

Combination of Lenses

When multiple lenses are used together, the effective focal length of the combination can be determined using the formula 1/ƒ_T = 1/ƒ_C + 1/ƒ_D. This relationship shows how the focal lengths of individual lenses interact, allowing for the design of optical systems with desired properties by combining different types of lenses.
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Related Practice
Textbook Question

Two 28.0-cm-focal-length converging lenses are placed 16.5 cm apart. An object is placed 35.0 cm in front of one lens.

(a) Where will the final image formed by the second lens be located?

(b) What is the total magnification?

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Textbook Question

(II) An aquarium filled with water has flat glass sides whose index of refraction is 1.51. A beam of light from outside the aquarium strikes the glass at a 43.5° angle to the perpendicular (Fig. 32–52). What is the angle of this light ray when it enters (a) the glass, and then (b) the water? (c) What would be the refracted angle if the ray entered the water directly?

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Textbook Question

"(II) Two plane mirrors meet at a 135° angle, Fig. 32–47. If light rays strike one mirror at 32° as shown, at what angle θ do they leave the second mirror?


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Textbook Question

(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 dT = di + do 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.

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Textbook Question

An object is placed 96.0 cm from a glass lens (n = 1.52) with one concave surface of radius 22.0 cm and one convex surface of radius 18.5 cm.

(a) Where is the final image?

(b) What is the magnification?

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Textbook Question

A diverging lens with ƒ = -36.5 cm is placed 14.0 cm behind a converging lens with ƒ = 20.0cm. Where will an object at infinity be focused?