Question

The photograph of a house is being occupied in an area of $1.75c{{m}^{2}}$ on a $35mm$ slide. The slide is being projected onto a screen, and the area of the house on the screen is $1.55{{m}^{2}}$. Calculate the linear magnification of the projector-screen arrangement.

Answer 1

Hint: Arial magnification can be found by taking the ratio of the area of the image to the area of the object. The square of the linear magnification is equivalent to the arial magnification also. This will help you in answering this question.

Complete step by step answer:
The area of the object has been mentioned in the question as,
${{A}_{o}}=1.75c{{m}^{2}}$
This has to be converted in terms of metres. So that we can write that,
\[{{A}_{o}}=1.75\times {{10}^{-4}}{{m}^{2}}\]
And the area of the image has been given as,
${{A}_{i}}=1.55{{m}^{2}}$
Arial magnification can be found by taking the ratio of the area of the image to the area of the object. This can be expressed in the equation as,
${{m}_{a}}=\dfrac{{{A}_{i}}}{{{A}_{o}}}$
Substituting the values in it will give,
${{m}_{a}}=\dfrac{1.55}{1.75\times {{10}^{-4}}}=8857$
As we all know, the square of the linear magnification is equivalent to the arial magnification. This can be written as an equation such that,
${{m}_{a}}={{m}_{l}}^{2}$
Where ${{m}_{l}}$ be the linear magnification. Rearranging the equation can be written as,
${{m}_{l}}=\sqrt{{{m}_{a}}}$
Substituting the values in it will give,
${{m}_{l}}=\sqrt{8857}=94.11$
Therefore the linear magnification has been obtained.

Note: Linear magnification is defined as the ratio of length of the image to that of the object determined in planes which are perpendicular to the optical axis. A negative value of linear magnification indicates that an inverted image is formed. Linear magnification is sometimes called lateral magnification. Areal magnification refers to the measure of enlargement of the area of an object. Children’s microscopes and hand lenses can be considered as the devices with areal magnification. Most other optical equipment are having linear magnifications.