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Chapter 5: Problem 30

We would like to place an object \(45\) \(\mathrm{cm}\) in front of a lens and have its image appear on a screen \(90\) \(\mathrm{cm}\) behind the lens. What must be the focal length of the appropriate positive lens?

Short Answer

Expert verified
The focal length of the lens must be 30 cm.

Step by step solution

01

Identify Known Variables

We are given the object distance, which is the distance between the object and the lens, denoted as \(d_o = 45\, \mathrm{cm}\). We are also given the image distance, which is the distance between the image and the lens, denoted as \(d_i = 90\, \mathrm{cm}\). We need to find the focal length of the lens, \(f\).
02

Use the Lens Formula

The lens formula relates the object distance \((d_o)\), the image distance \((d_i)\), and the focal length \((f)\) as follows: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\).
03

Substitute Values into the Lens Formula

Substitute \(d_o = 45\, \mathrm{cm}\) and \(d_i = 90\, \mathrm{cm}\) into the lens formula: \(\frac{1}{f} = \frac{1}{45} + \frac{1}{90}\).
04

Calculate the Right-Hand Side

Calculate \(\frac{1}{45} + \frac{1}{90}\): First, find a common denominator, which is 90. Therefore, \(\frac{1}{45} = \frac{2}{90}\). Now, add \(\frac{2}{90} + \frac{1}{90} = \frac{3}{90}\).
05

Solve for Focal Length \(f\)

We found that \(\frac{1}{f} = \frac{3}{90}\), which simplifies to \(\frac{1}{f} = \frac{1}{30}\). Therefore, by taking the reciprocal, \(f = 30\, \mathrm{cm}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lens Formula
When working with optics, particularly lenses, the lens formula is a fundamental equation that helps us determine key characteristics of a lens system. This formula is essential for calculating one of the three primary measurements in lens problems: the object distance \(d_o\), the image distance \(d_i\), and the focal length \(f\). The lens formula is expressed as:
  • \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)
This equation indicates the reciprocal relationship among the focal length, object distance, and image distance.
To use the lens formula effectively, you usually need two known values to solve for the unknown third value.
The symmetry in this formula allows us to calculate the lens characteristics for both converging (positive) and diverging (negative) lenses.
Object Distance
Object distance, denoted by \(d_o\), is a crucial parameter in any experiment involving lenses. It refers to how far the object is placed from the lens.
In optical calculations and experiments, accurately knowing this distance is critical for obtaining the correct focal length or image distance.
Recollect that an object placed further from the lens will generally produce a smaller image, assuming the lens and image distances remain constant.
  • For our exercise, this distance is 45 cm in front of the lens.
This value helps set the stage for using the lens formula effectively as a given input to find the focal length.
Image Distance
The image distance \(d_i\) is another vital metric in lens calculations, indicating the distance between the lens and the projected image.
Physically, this distance is the separation on the opposite side of the lens from the object, and it can tell us much about the type and nature of the image formed.
For example, a real image will fall on the same side as the object for a positive lens and can vary in size, depending on \(d_i\).
  • In our specific case, the image distance is 90 cm, placed behind the lens.
Having this value, along with the object distance, allows us to apply the lens formula to determine the focal length.
Positive Lens
A positive lens, also known as a converging lens, is designed to focus incoming parallel rays to a single point, known as the focal point.
These lenses are thicker at the center than at the edges and are commonly used in a variety of optical devices, such as cameras and eyeglasses.
The focal length of a positive lens is positive, reflecting its ability to converge light rays to form a real or virtual image. In problems involving a positive lens, the lens formula is applied in the same way, but the signs of the measurements help determine the nature of the image formed.
  • The problem we solved uses a positive lens to determine a 30 cm focal length.
This characteristic is critical to understanding how lenses form images and how to correctly apply optical principles in solving lens-related problems.

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Most popular questions from this chapter

A refracting astronomical telescope has an objective lens \(50 \mathrm{mm}\) in diameter. Given that the instrument has a magnification of \(10 \times,\) determine the diameter of the eye -beam (the cylinder of light impinging on the eye). Under conditions of darkness the acclimated human eye has a pupil diameter of about \(8\) \(\mathrm{mm}\).

Two positive lenses with focal lengths of \(0.30\) \(\mathrm{m}\) and \(0.50\) \(\mathrm{m}\) are separated by a distance of \(0.20\) \(\mathrm{m}\). A small butterfly rests on the central axis \(0.50 \mathrm{m}\) in front of the first lens. Locate the resulting image with respect to the second lens.

Given a fiber with a core diameter of \(50 \mu \mathrm{m}\) and \(n_{c}=1.482\) and \(n_{f}=1.500,\) determine the number of modes it sustains when the fiber is illuminated by an LED emitting at a central wavelength of \(0.85 \mu \mathrm{m}\).

Compute the focal length in air of a thin biconvex lens \(\left(n_{l}=1.5\right)\) having radii of 20 and \(40 \mathrm{cm} .\) Locate and describe the image of an object \(40 \mathrm{cm}\) from the lens.

A thin convex lens \(L\) is positioned midway between two diaphragms: \(D_{1}, 4.0 \mathrm{cm}\) to its left, and \(D_{2}, 4.0 \mathrm{cm}\) to its right. The lens has a diameter of \(12 \mathrm{cm}\) and a focal length of \(12 \mathrm{cm} .\) The holes in \(D_{1}\) and \(D_{2}\) have diameters of \(12 \mathrm{cm}\) and \(8.0 \mathrm{cm},\) respectively. An axial object point is \(20 \mathrm{cm}\) to the left of \(D_{1} .\) (a) What is the image of \(D_{1}\) in the object space (i.e., as imaged by any lens to its left with light traveling left \() ?(\mathrm{b})\) What is the image of \(L\) in the object space? (c) What is the image of \(D_{2}\) in the object space? Give the size and location of that aperture's image. (d) Locate the entrance pupil and the aperture stop.
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