Skip to main content
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 112
Chapter 32, Problem 112

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?

Verified step by step guidance
1
Step 1: Begin by recalling the lens formula: 1f=1dₒ+1dᵢ, where f is the focal length, dₒ is the object distance, and dᵢ is the image distance. Additionally, magnification m is defined as m=-dᵢdₒ. These two equations will be key to solving the problem.
Step 2: Substitute the expression for dₒ from the lens formula into the magnification formula. Rearrange the lens formula to express dₒ in terms of f and dᵢ: dₒ=dᵢdᵢ-f. Then substitute this into the magnification formula: m=-dᵢdᵢdᵢ-f. Simplify the expression to show that m is a linear function of dᵢ.
Step 3: Simplify the magnification formula further to express m in the form of a straight-line equation: m=-dᵢf+1. This equation is in the form y = mx + b, where the slope is -1f and the y-intercept is 1.
Step 4: Using the provided experimental data, plot m vs. dᵢ on a graph. Fit a straight line to the data points and determine the slope and y-intercept of the line. The slope should correspond to -1f, and the y-intercept should be 1, as derived theoretically.
Step 5: Use the slope of the line to calculate the focal length f of the lens. Verify whether the y-intercept of the graph matches the expected theoretical value of 1. If discrepancies exist, consider experimental errors or uncertainties in the measurements.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Lens Formula

The lens formula relates the object distance (dₒ), image distance (dᵢ), and focal length (f) of a lens through the equation 1/f = 1/dₒ + 1/dᵢ. This formula is fundamental in optics, allowing us to understand how light converges through a lens and how the position of the object affects the image formed.
Recommended video:
05:38
Lens Maker Equation

Magnification

Magnification (m) is defined as the ratio of the height of the image (hᵢ) to the height of the object (hₒ), and it can also be expressed as m = -dᵢ/dₒ. This negative sign indicates that the image is inverted. Understanding magnification is crucial for analyzing how the size of the image changes with respect to the object distance.
Recommended video:
09:03
Mirror Equation

Graphing Relationships

In this experiment, plotting magnification (m) against image distance (dᵢ) should yield a linear relationship, as derived from the lens formula and magnification equations. The slope of this line can be used to determine the focal length of the lens, while the y-intercept provides insight into the theoretical behavior of the system, helping to validate the experimental results.
Recommended video:
Guided course
07:32
Graphing Position, Velocity, and Acceleration Graphs
Related Practice
Textbook Question

An astronomical telescope, Fig. 33–36, produces an inverted image. One way to make a telescope that produces an upright image is to insert a third lens between the objective and the eyepiece, Fig. 33–39b. To have the same magnification, the non-inverting telescope will be longer. Suppose lenses of focal length 150 cm, 1.5 cm, and 10 cm are available. Where should these three lenses be placed to make a non-inverting telescope with magnification 100x?

1
views
Textbook Question

A physicist lost in the mountains tries to make a telescope using the lenses from his reading glasses. They have powers of +2.0 D and +4.5 D, respectively.

(a) What maximum magnification telescope is possible?

(b) Which lens should be used as the eyepiece?

1
views
Textbook Question

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

1
views