Question

A thin pencil of length \( f/4 \) is placed coinciding with the principal axis of a mirror of focal length \( f \). The image of the pencil is real and enlarged, just touches the pencil. Calculate the magnification produced by the mirror.

Hint:

The magnification produced by a mirror can be found using the mirror equation and the condition that the image just touches the object.

Answer 1

The magnification \( m \) produced by a mirror is given by the formula: \[ m = - \frac{v}{u} \] Where: 
\( v \) is the image distance,
\( u \) is the object distance.

Given that the image is real and enlarged, the image distance is positive, and the object distance is negative. The condition that the image just touches the pencil means the image and object distances add up to the focal length. Therefore: \[ v + u = f \] Also, the relationship between the focal length \( f \), object distance \( u \), and image distance \( v \) for a mirror is given by the mirror equation: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] From these two equations, we can calculate the magnification produced by the mirror.