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Chapter 23: Problem 73

An object \(8.0 \mathrm{cm}\) high forms a virtual image \(3.5 \mathrm{cm}\) high located \(4.0 \mathrm{cm}\) behind a mirror. (a) Find the object distance. (b) Describe the mirror: is it plane, convex, or concave? (c) What are its focal length and radius of curvature?

Short Answer

Expert verified
What are the object distance, focal length, and radius of curvature? This is a convex mirror with an object distance of 9.143 cm, a focal length of 12.308 cm, and a radius of curvature of 24.615 cm.

Step by step solution

01

Calculate the Magnification Ratio

The magnification ratio (M) is the ratio of the image height (hi) to the object height (ho). We can calculate it by: $$ M = \frac{h_i}{h_o} $$ For this problem, \(h_o = 8.0\,\text{cm}\) and \(h_i = 3.5\,\text{cm}\). Plugging the values we get: $$ M = \frac{3.5}{8.0} $$
02

Determine the type of mirror

If the magnification ratio is positive, then the mirror is a virtual mirror and consequently, it is a convex mirror. If it is negative, then it is a concave mirror. In our calculation: $$ M = \frac{3.5}{8.0} > 0 $$ Since the magnification ratio is positive, the mirror is a convex mirror.
03

Calculate the object distance (do)

Since we know the magnification ratio (M), we can rewrite the magnification formula for a mirror as: $$ M = \frac{-d_i}{d_o} $$ We are given the image distance (\(d_i = -4.0\,\text{cm}\), negative because it is behind the mirror). So we can solve for the object distance (do): $$ d_o = -\frac{d_i}{M} $$ Plugging the values, we get: $$ d_o = -\frac{-4.0}{\frac{3.5}{8.0}} $$
04

Calculate the focal length (f) using the mirror equation

The mirror equation can be written as: $$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $$ We have calculated \(d_o\) and we are given \(d_i\). So we can solve for the focal length (f): $$ f = \frac{1}{\frac{1}{d_o} + \frac{1}{d_i}} $$ Plugging the values of \(d_o\) and \(d_i\), we can find the value of \(f\).
05

Calculate the radius of curvature (R)

The radius of curvature (R) is twice the focal length (f) for a mirror. So we can calculate it as: $$ R = 2f $$ Plugging in the value of focal length (f) that we calculated in step 4, we can find the radius of curvature. Now, you have the object distance (d_o), the type of mirror (convex), the focal length (f), and the radius of curvature (R).

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