Question

A lens forms a virtual, diminished image of an object placed at 2 m from it. The size of the image is half of the object. Which one of the following statements is correct regarding the nature and focal length of the lens?
a) Concave, $|f| = 1m$
b) Convex, $|f| = 1m$
c) Concave, $|f| = 2m$
d) Convex, $|f| = 2m$

Answer 1

Hint: A convex lens can form both real and virtual images which are magnified in size and inverted compared to the object. A concave lens always forms virtual images which are erect and diminished in size.

Formula used:
Lens formula is given as:
$\dfrac{1}{f} = \dfrac{1}{{\text{v}}} - \dfrac{1}{u}$
where f is used to represent the focal length of the lens, while u and v represent the distance of the object and its image respectively from the lens.
Magnification is given as
$m = \dfrac{{\text{v}}}{u}$

Detailed step by step solution:
There are two types of spherical lens: Concave lens and convex lens.
Concave lens is also called a diverging lens because it diverges the light rays passing through it. The images formed are virtual, erect and diminished.
A convex lens is called a converging lens as it converges the light rays passing through it. The images formed can be real or virtual, magnified and inverted.
We are given that $u = - 2m$ and magnification $m = 0.5$
$\therefore \dfrac{{\text{v}}}{u} = 0.5 \\\ \Rightarrow {\text{v}} = 0.5u = 0.5 \times \left( { - 2} \right) = - 1 \\\$
Therefore the image is formed on the left side of the lens. Now using the lens formula, we get
$\dfrac{1}{f} = \dfrac{1}{{\text{v}}} - \dfrac{1}{u} \\\ \Rightarrow \dfrac{1}{f} = \dfrac{1}{{ - 1}} - \dfrac{1}{{ - 2}} = - 1 + \dfrac{1}{2} = - \dfrac{1}{2} \\\ \Rightarrow f = - 2m \\\$
Since, a concave lens has negative focus, therefore, the correct answer is option C.

Note: 1. All distances to the left of the lens are taken to be negative while all distances to the right of the lens are taken to be negative.
2. The focus is defined on the basis of curvature of the front circle of the lens. A concave lens has negative focus because it lies on the left side of the lens while a convex lens has positive focus because it lies on the right side of the lens.