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Chapter 18: Problem 132

Assuming that we have taken the origin at the same place as in previous question. A point object is placed on the principal axis \(18 \mathrm{~cm}\) from the origin in front of a convex mirror of focal length \(6 \mathrm{~cm}\). The distance of image from the pole of mirror is (A) \(6 \mathrm{~cm}\) (B) \(10 \mathrm{~cm}\) (C) \(25 \mathrm{~cm}\) (D) \(4 \mathrm{~cm}\)

Short Answer

Expert verified
The location of the image from the pole of the mirror is 9 cm in the direction opposite to the incident light, so option (A) is incorrect, but none of the options provided match the solution.

Step by step solution

01

Understand the mirror equation

The mirror equation can be computed by taking the reciprocal of both sides in the equation, which gives \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\). Here, \(f\) is the mirror's focal length, \(v\) is the image's location, and \(u\) is the object's location.
02

Apply the mirror sign conventions

In the case of mirrors, the sign conventions dictate that any distance measured in the same direction as the incident light is positive, and distances measured in the opposite direction are negative. For a convex mirror, the focal length \(f\) is always negative. Similarly, for object distances \(u\) measured in the same direction as light, the convention denotes it as negative. As such in this case, \(u = -18 \mathrm{~cm}\) and \(f = -6 \mathrm{~cm}\).
03

Use the mirror equation to find the image's location

We can now substitute the object's location and the mirror's focal length into the mirror equation, and solve for \(v\). Upon substitution, we get \(\frac{1}{v} = \frac{1}{-6} - \frac{1}{-18}\). Solving for \(v\), we find that \(v = -9 \mathrm{~cm}\). The negative sign indicates that the image is located in the direction opposite to the incident light.

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