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Ch. 32 - Light: Reflection and Refraction
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. 32 - Light: Reflection and RefractionProblem 32
Chapter 31, Problem 32

An object is placed a distance r in front of a wall, where r exactly equals the radius of curvature of a certain concave mirror. At what distance from the wall should this mirror be placed so that a real image of the object is formed on the wall? What is the lateral magnification of the image?

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Understand the problem: We are tasked with determining the distance at which a concave mirror should be placed from a wall so that a real image of an object is formed on the wall. Additionally, we need to calculate the lateral magnification of the image. The key information provided is that the object is placed at a distance r from the wall, where r equals the radius of curvature of the mirror.
Recall the mirror equation: The mirror equation is given by: \(\frac{1}{f}\) = \(\frac{1}{d_o}\) + \(\frac{1}{d_i}\), where f is the focal length of the mirror, d_o is the object distance, and d_i is the image distance. For a concave mirror, the focal length is related to the radius of curvature by f = \(\frac{r}{2}\).
Substitute the given information: Since the object is placed at a distance r from the wall, the object distance d_o is the distance between the mirror and the object. Let the distance between the mirror and the wall be x. Then, the object distance is d_o = r - x, and the image distance is d_i = x (since the image is formed on the wall).
Rewrite the mirror equation: Substitute d_o = r - x, d_i = x, and f = \(\frac{r}{2}\) into the mirror equation: \(\frac{1}{\frac{r}{2}\)} = \(\frac{1}{r - x}\) + \(\frac{1}{x}\). Simplify this equation to solve for x, the distance between the mirror and the wall.
Determine the lateral magnification: The lateral magnification M is given by M = -\(\frac{d_i}{d_o}\). Substitute d_i = x and d_o = r - x into this formula to calculate the magnification. Simplify the expression to find the magnification in terms of r and x.

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Key Concepts

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

Radius of Curvature

The radius of curvature of a mirror is the distance from the mirror's surface to its center of curvature, which is the point where the mirror's surface would meet if extended. For concave mirrors, this radius is crucial in determining the focal point, which is located halfway between the center of curvature and the mirror's surface. Understanding this concept helps in analyzing how light rays reflect off the mirror and where images are formed.
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Mirror Formula

The mirror formula relates the object distance (u), image distance (v), and the focal length (f) of a mirror, expressed as 1/f = 1/v + 1/u. For concave mirrors, the focal length is negative, and this formula is essential for calculating where the image will form based on the object's position. By applying this formula, one can determine the necessary distance from the wall for the mirror to create a real image on the wall.
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Lateral Magnification

Lateral magnification (M) is the ratio of the height of the image (h') to the height of the object (h), and it can also be expressed as M = -v/u, where v is the image distance and u is the object distance. This concept indicates how much larger or smaller the image appears compared to the object. Understanding lateral magnification is important for interpreting the size and orientation of the image formed by the mirror.
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Related Practice
Textbook Question

(II) In searching the bottom of a pool at night, a watchman shines a narrow beam of light from his flashlight, 1.3 m above the water level, onto the surface of the water at a point 2.8 m from his foot at the edge of the pool (Fig. 32–53). Where does the spot of light hit the bottom of the pool which is 2.1 m deep? Measure from the bottom of the wall beneath his foot.


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Textbook Question

A light beam strikes a 2.5-cm-thick piece of plastic with a refractive index of 1.62 at a 45° angle. The plastic is on top of a 3.8-cm-thick piece of glass for which n = 1.47. What is the distance D in Fig. 32–51?

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Textbook Question

When walking toward a concave mirror you notice that your image flips at a distance of 0.80 m from the mirror. What is the radius of curvature of the mirror? [Hint: Carefully examine Section 32–4.]

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Textbook Question

A shaving or makeup mirror is designed to magnify your face by a factor of 1.8 (when compared to a flat mirror) when your face is placed 20.0 cm in front of it.

(a) What type of mirror is it?

(b) Describe the type of image that it makes of your face.

(c) Calculate the required radius of curvature for the mirror.

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Textbook Question

A flashlight beam strikes the surface of a pane of glass (n = 1.56) at a 69° angle to the normal. What is the angle of refraction?

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Textbook Question

Let the focal length of a convex mirror be written as ƒ = ―|ƒ|. Show that the lateral magnification m of an object a distance dₒ from this mirror is given by m = |ƒ| / (dₒ +|ƒ| ). Based on this relation, explain why your nose looks bigger than the rest of your face when looking into a convex mirror.

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