Question

A myopic person has a power of $- 1.25$ Dioptre. What is the focal length and the nature of the lens?
A. $50\;cm$ and convex lens
B. $80\;cm$ and convex lens
C. $50\;cm$ and concave lens
D. $80\;cm$ and concave lens

Answer 1

Myopia is a condition called near-sightedness or short sightedness which needs to be corrected by using a specific type of lens material. The formula for the power of a lens in terms of its focal length is applied in order to find the required answer. In order to find the nature of the lens, the properties of lenses and their respective rules and sign conventions must be applied.

Complete step by step answer:
The problem revolves around the concept of lenses and a common defect of the eye known as myopia. In order to find the focal length of the lens used and its nature we first need to understand the concept of power in lenses.

Power is basically the measure of the degree or the extent to which the lens can converge or diverge the light rays falling on it. This can be measured by taking into consideration the focal length of the lens. The focal length is the distance a focus from the lens’ optical center. It is the point where all the beams of rays converge or diverge together.

The power is measured in terms of the angle formed when the light rays bend after hitting the lens material. Hence, the power in lenses is defined as the tangent of the angle by which it converges or diverges a beam of light incident on it at a unit distance from its optical center.

When the focal length of the lens is said to be smaller then it will be seen that its ability to bend light rays is more and hence the angle formed will be more and the power of the lens will in-turn be more. Similarly, when the focal length is said to be larger then it will be seen that its ability to bend light rays is less and hence the angle formed will be less and the power of the lens is less. Thus, from this it can be understood that power and focal length is inversely related. Thus the formula for the power of lenses is given by the equation:
$P = \dfrac{1}{f}$
We are asked to find the focal length of the lens and hence by rearranging the terms we get:
$f = \dfrac{1}{P}$ ---------($1$)

Let us now understand the term myopia. A normal eye can see objects between a distance of $25cm$and infinity but over time the vision of the eye may become defective and hence the objects that are to be viewed become unclear or blurry. One of the common types of defects of the eye is a condition called myopia. Myopia is a condition wherein objects at a distance cannot be viewed clearly. It is defined as the vision defect in which a person can see nearby objects clearly but cannot see the distant beyond a certain point.

Let us now extract the data given in the question. We are given the power of the lens to be $- 1.25$ Dioptre. This unit Dioptre is actually defined as the unit of power of the lens when the focal length is said to be in one meter. Hence by substituting the value of power in equation ($1$) we get the focal length in terms of meters. Hence we get:
$f = \dfrac{1}{{ - 1.25}}$
$\Rightarrow f = - \dfrac{1}{{1.25}}$
On solving the above equation we get the focal length to be:
$f = - 0.8\,m$

We now convert meters into centimeters since the focal length of lens to correct myopia is measured in terms of $cm$.
We know that.
$1m = 100\,cm$
Hence, $- 0.8\,m = - 0.8 \times 100\,cm$
$\therefore f = - 80\;cm$
Thus, this is the focal length of the lens that is used.Now, we come to deducing the nature of the lens. Nature of the lens basically means the type of lens that is used; that is, either convex or concave. Here, we can observe that the focal length is negative and hence the nature of the lens used will be concave in nature in order to correct the defect produced by the eye.

This is in accordance with the sign convention for concave lenses.A concave lens is used to correct a myopic because the image formed by myopic eye is formed in-front of the retina instead of forming on the retina so when a concave lens is used then the light rays are diverged and the image is formed correctly at the retina and hence a concave lens is used.

Hence, the correct answer is option D.

Note: A common error that can be made is that sign conventions can be mixed up and correspondingly the nature of the lens can be mistaken to be a convex lens. A concave lens is thinner at the center and bulged at the ends in stark contrast to convex lenses. Its property is to diverge the parallel light rays incident on it and hence it is known as a diverging lens as well.