Question

A lens forms a virtual, erect and diminished image of an object placed at a distance of $2m$ from it. The size of the image formed is half of that of the object. Which of the following statement is correct regarding the nature and focal length of the lens?

• A. Concave lens, $f=2m$
• B. Convex lens, $f=2m$
• C. Concave lens, $f=-2m$
• D. Convex lens, $f=-2m$

Answer 1

The correct option is C

Concave lens, $f=-2m$

Let $u$ be the object distance, $v$ be the image distance,
and $f$ be the focal length.
Given:
As per sign conventions, $u=-2m$
Magnification $m=\frac{1}{2}$ (Since size of image is half of that of object)
Magnification in lenses $m=\frac{v}{u}$
$\frac{1}{2}=\frac{v}{u}$
We get $v=\frac{u}{2}$
$\Rightarrow v=\frac{-2}{2}=-1m$
Using lens formula ,
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$

$\frac{1}{-1}-\frac{1}{-2}=\frac{1}{f}$

$\frac{1}{2}-1=\frac{1}{f}$

$f=-2m$
Here negative sign indicates that the lens is a concave lens.