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 \( m \) of a mirror is calculated using the formula: \[ m = - \frac{v}{u} \] In this formula:
\( v \) represents the image distance,
\( u \) represents the object distance.

For a real and enlarged image, the image distance \( v \) is positive, and the object distance \( u \) is negative. The condition that the image precisely touches the pencil implies that the sum of the image and object distances equals the focal length. Consequently: \[ v + u = f \] Additionally, the relationship between the focal length \( f \), object distance \( u \), and image distance \( v \) for a mirror is defined by the mirror equation: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] These two equations can be used to determine the magnification produced by the mirror.