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.

Answer 1

Given, 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, enlarged, says the mirror must be a concave mirror with the object placed between f and 2f to ensure that its image forms beyond 2f, and only touches suggests that the image coincides with one end of the pencil.

Assume that the object is placed u from the mirror, creating an image at v. The magnification is given by:

m = `(-v)/u`

We assume the object is placed at u = 1.5 f, or `((3 f)/2)`, and it is an enlarged and real image.

By the mirror formula:

`1/f = 1/u + 1/v`

Substituting, u = `(3 f)/2`

`1/f = 1/((3 f)/2) + 1/v`

`1/f = 2/(3 f) + 1/v`

`1/f - 2/(3 f) = 1/v`

`((3 - 2))/(3 f) = 1/v`

`1/(3 f) = 1/v`

v = 3f

Magnification (m) = `(-v)/u`

= `(-3 f)/((3 f)/2)`

= −2

Since magnification is negative, the image is inverted and twice the size of the object.