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Chapter 23: Problem 50

An object is placed in front of a convex mirror with a 25.0 -cm radius of curvature. A virtual image half the size of the object is formed. At what distance is the object from the mirror? Draw a ray diagram to illustrate.

Short Answer

Expert verified
Answer: The object is 33.34 cm away from the mirror, and the ray diagram shows a virtual, upright, and smaller image formed behind the mirror.

Step by step solution

01

Determine the focal length of the convex mirror

Using the radius of curvature (R) provided, we can calculate the focal length (f) using the formula f = R/2. f = (25.0 cm)/2 f = 12.5 cm Since it is a convex mirror, the focal length is negative: f = -12.5 cm.
02

Determine the magnification

Since the virtual image is half the size of the object, the magnification (m) is -0.5 (the negative sign indicates that the image is virtual).
03

Use the mirror and magnification formulas

The mirror formula is given by: 1/f = 1/u + 1/v The magnification formula (with -0.5 magnification) is: m = -v/u -0.5 = -v/u We can rewrite the magnification formula as: u = 2v Now, substitute these values into the mirror formula: 1/f = 1/(2v) + 1/v Substitute the value of f: 1/(-12.5 cm) = 1/(2v) + 1/v
04

Solve the equation for the distance of the object, u

We can solve for the distance v using algebra, and then find the distance u: 1/(-12.5 cm) = 1/(2v) + 1/v 1/v - 1/(-12.5 cm) = 1/(2v) v = 1/[(1/(-12.5 cm))+(1/(-25.0 cm))] v 鈮� -16.67 cm Now we can find the distance of the object, u, by using our previously found relationship, u = 2v: u = 2 * (-16.67 cm) u = -33.34 cm Because the distance u is negative, this confirms that the image formed by the convex mirror is virtual.
05

Draw a ray diagram

To illustrate the situation, draw a ray diagram, including the following components: 1. Draw a horizontal line to represent the principal axis. 2. Place the convex mirror along the principal axis, showing the center of curvature (C) and the focal point (F). 3. Place the object in front of the mirror with its tip on the principal axis. 4. Draw a ray from the tip of the object to the mirror parallel to the principal axis. After reflecting, extend the ray on the other side of the mirror from the focal point. 5. Draw another ray from the tip of the object straight to the center of the mirror. This ray will reflect back on itself. 6. Extend the reflected rays behind the mirror to the point where they appear to intersect. This point will be the location of the virtual image. The image should be smaller and closer to the mirror than the object. Image distance (v) is -16.67 cm and object distance (u) is -33.34 cm, which represent the virtual and upright nature of the image.

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