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Chapter 33: Problem 91

An instructor wants to use a lens to project a real image of a light bulb onto a screen \(1.71 \mathrm{~m}\) from the bulb. In order to get the image to be twice as large as the bulb, what focal length lens will be needed?

Short Answer

Expert verified
Answer: The focal length of the lens needed is approximately 1.14 meters.

Step by step solution

01

Define Variables

Let's denote the focal length as \(f\), the object distance as \(d_o\), the image distance as \(d_i\), and the magnification as \(m\). In this problem, we have: \(d_o = 1.71m\) \(m = 2\)
02

Using the Lens Maker's Equation

The lens maker's equation is given by: \(\frac{1}{f} = \frac{1}{d_i} + \frac{1}{d_o}\), where \(f\) is the focal length, \(d_i\) is the image distance, and \(d_o\) is the object distance. We want to find the value of \(f\).
03

Using the Magnification Equation

The magnification equation is given by: \(m = \frac{d_i}{d_o}\). Since we want the magnification to be \(2\), and the object distance is \(1.71m\), we can find the image distance \(d_i\): \(m = 2 = \frac{d_i}{1.71}\) \(d_i = 2 \times 1.71 = 3.42m\)
04

Substituting Values into the Lens Maker's Equation

Now that we have the values for \(d_o\) and \(d_i\), we can plug them into the lens maker's equation to find the focal length \(f\): \(\frac{1}{f} = \frac{1}{3.42} + \frac{1}{1.71}\) \(\frac{1}{f} = 0.2923 + 0.5848\) \(\frac{1}{f} = 0.8771\)
05

Solving for the Focal Length

Now, we'll solve for the focal length by taking the reciprocal of the resulting value: \(f = \frac{1}{0.8771}\) \(f \approx 1.14m\) The focal length of the lens needed is approximately \(1.14m\).

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Most popular questions from this chapter

Two refracting telescopes are used to look at craters on the Moon. The objective focal length of both telescopes is \(95.0 \mathrm{~cm}\) and the eyepiece focal length of both telescopes is \(3.80 \mathrm{~cm} .\) The telescopes are identical except for the diameter of the lenses. Telescope A has an objective diameter of \(10.0 \mathrm{~cm}\) while the lenses of telescope \(\mathrm{B}\) are scaled up by a factor of two, so that its objective diameter is \(20.0 \mathrm{~cm}\). a) What are the angular magnifications of telescopes \(A\) and \(B\) ? b) Do the images produced by the telescopes have the same brightness? If not, which is brighter and by how much?

What type of lens is a magnifying glass? a) converging d) cylindrical b) diverging e) plain c) spherical

Several small drops of paint (less than \(1 \mathrm{~mm}\) in diameter) splatter on a painter's eyeglasses, which are approximately \(2 \mathrm{~cm}\) in front of the painter's eyes. Do the dots appear in what the painter sees? How do the dots affect what the painter sees?

For a microscope to work as intended, the separation between the objective lens and the eyepiece must be such that the intermediate image produced by the objective lens will occur at a distance (as measured from the optical center of the eyepiece) a) slightly larger than the focal length. b) slightly smaller than the focal length. c) equal to the focal length. d) The position of the intermediate image is irrelevant.

An unknown lens forms an image of an object that is \(24 \mathrm{~cm}\) away from the lens, inverted, and a factor of 4 larger in size than the object. Where is the object located? a) \(6 \mathrm{~cm}\) from the lens on the same side of the lens b) \(6 \mathrm{~cm}\) from the lens on the other side of the lens c) \(96 \mathrm{~cm}\) from the lens on the same side of the lens d) \(96 \mathrm{~cm}\) from the lens on the other side of the lens e) No object could have formed this image.
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