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Chapter 31: Problem 40

You're asked to design a concave mirror that will produce a virtual image, enlarged 1.8 times, of an object \(22 \mathrm{cm}\) from the mirror. What do you specify for the mirror's curvature radius?

Short Answer

Expert verified
The radius of curvature for the mirror needed for this task is -31.8 cm.

Step by step solution

01

Calculation of Image Distance

The magnification \(M\) of a mirror is negative for a real image and positive for a virtual image. Given that the image is virtual and is magnified 1.8 times, the magnification \(M=-1.8\). The object distance \(o = -22 cm\). Substituting these values in the magnification formula \(M=\frac{-i}{o}\), we get: \(-1.8 = \frac{-i}{-22}\). From this we can solve for \(i\), which is -39.6 cm.
02

Calculation of Focal Length

Now, using the mirror formula \(\frac{1}{f} = \frac{1}{o} + \frac{1}{i}\), you can substitute the values \(o = -22 cm\) and \(i = -39.6 cm\) to find: \(\frac{1}{f} = \frac{1}{-22} + \frac{1}{-39.6}\). Now, solve this equation to determine the value of \(f\), the focal length. After calculating, (f) is found to be -15.9cm.
03

Calculation of Mirror's Curvature Radius

The radius of curvature \(R\) of a mirror is twice its focal length. Given that the focal length of the mirror, as found in the previous step, is \(f = -15.9cm\), you get the radius \(R = 2f\). Substituting for \(f\) gives \(R = 2(-15.9cm) = -31.8cm\). Thus, the mirror's curvature radius is -31.8 cm. The negative sign indicates that the mirror is concave.

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Most popular questions from this chapter

Does a fish in a spherical bowl appear larger or smaller than it actually is?

A candle is on the axis of a \(15-\mathrm{cm}\) -focal-length concave mirror, \(36 \mathrm{cm}\) from the mirror. (a) Where is its image? (b) How do the image and object sizes compare? (c) Is the image real or virtual?

For what refractive index would the focal length of a plano-convex lens be equal to the curvature radius of its one curved surface?

If you're handed a converging lens, what can you do to estimate its focal length quickly?

LCD projectors commonly used for computer and video projection create an image on a small LCD display (see Application on page 369 ). The display is mounted before a lens and illuminated from behind. In a projector using a \(7.50-\mathrm{cm}\) -focal-length convex lens, where should the LCD display be located so the projected image is focused on a screen \(6.30 \mathrm{m}\) from the lens?
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