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Chapter 2: Problem 92

An object viewed with the naked eye subtends a \(2^{\circ}\) angle. If you view the object through a \(10 \times\) magnifying glass, what angle is subtended by the image formed on your retina?

Short Answer

Expert verified
The angle subtended by the image formed on the retina when viewed through a \(10 \times\) magnifying glass is \(20^{\circ}\).

Step by step solution

01

Identify the given quantities

We are given the following: 1. Angle subtended by the object when viewed with the naked eye, which is \(2^{\circ}\). 2. Magnification factor of the magnifying glass, which is \(10 \times\).
02

Use the formula for magnification

The formula for magnification is given by: Magnification = \(\frac{\text{Angle subtended by image}}{\text{Angle subtended by object}}\) Let the angle subtended by the image formed on the retina when viewed through the magnifying glass be x. Then, the formula becomes: \(10 = \frac{x}{2}\)
03

Solve for the new angle subtended by the image

To solve for x, we can multiply both sides of the equation by the angle subtended by the object (2): \(10 \times 2 = x\) Now, simplify the equation: \(20 = x\)
04

State the final answer

The angle subtended by the image formed on the retina when viewed through a \(10 \times\) magnifying glass is \(20^{\circ}\).

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Key Concepts

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

Angle Subtended
Understanding the concept of the angle subtended by an object is foundational in geometric optics. This angle is basically the angle formed at the viewer's eye by lines drawn from the top and bottom of the object. Imagine looking at a building nearby; the angle your eyesight makes from the roof to the base as seen from your position is the angle subtended.

The size of this angle depends on the object's size and the distance from the observer. If the object appears larger or comes closer, the angle increases, and vice versa. This concept is linked to how we perceive the size of objects; the larger the angle subtended, the bigger the object appears to us.
Magnifying Glass
A magnifying glass, a common tool for optical magnification, works by creating a magnified virtual image of an object so it subtends a larger angle at the eye's lens than if the object were viewed with the naked eye.

Constructed with a convex lens, a magnifying glass bends (refracts) light rays so that they converge at a point. When you place an object within the focal length of the magnifying glass, you can see an upright and enlarged image of that object, making it easier to view fine details.
Optical Magnification
Optical magnification is a key concept in optics, defined as the process of enlarging the appearance of an object via an optical instrument. In mathematical terms, magnification can be expressed as the ratio of the angle subtended by the image to the angle subtended by the object itself, as demonstrated in the given exercise.

When an object is magnified, such as with a magnifying glass, the larger angle at which the image is seen results in what we perceive as a larger or closer object. This is crucial in various applications like microscopy, photography, and even in simple tasks like reading small print.
Geometric Optics
Geometric optics is a branch of optics that describes light propagation in terms of rays. The ray approximation is used to explain how light travels through mediums and interacts with surfaces, like reflection and refraction.

In the case of a magnifying glass or lenses, geometric optics helps describe how light is focused to a point, forming images. It simplifies the understanding of complex phenomena and provides a basis for designing and using optical devices in everyday life and in scientific equipment.

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Most popular questions from this chapter

What are the differences between real and virtual images? How can you tell (by looking) whether an image formed by a single lens or mirror is real or virtual?

You photograph a 2.0-m-tall person with a camera that has a \(5.0 \mathrm{cm}\) -focal length lens. The image on the film must be no more than \(2.0 \mathrm{cm}\) high. (a) What is the closest distance the person can stand to the lens? (b) For this distance, what should be the distance from the lens to the film?

A certain slide projector has a 100 mm-focal length lens. (a) How far away is the screen if a slide is placed \(103 \mathrm{mm}\) from the lens and produces a sharp image? (b) If the slide is 24.0 by \(36.0 \mathrm{mm}\), what are the dimensions of the image? Explicitly show how you follow the steps in the Problem-Solving Strategy: Lenses.

Suppose a 200 mm-focal length telephoto lens is being used to photograph mountains \(10.0 \mathrm{km}\) away. (a) Where is the image? (b) What is the height of the image of a \(1000 \mathrm{m}\) high cliff on one of the mountains?

What is the focal length of a magnifying glass that produces a magnification of 3.00 when held \(5.00 \mathrm{cm}\) from an object, such as a rare coin?
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