Question

An object when placed in front of a plano-convex lens forms a real image at a distance of $10m$ behind the lens. The image formed is one-fourth the size of the object. The wavelength of light inside the lens is $\frac{3}{4}$ times the wavelength in free space. The radius of the curved surface of the lens is ______ $cm$.

Answer 1

Answer: 2.67

$\mu =\frac{{\lambda }_{\text{vacuum }}}{{\lambda }_{\text{med }}}=\frac{4}{3}$
Given, $v=+10m,m=\frac{-1}{4}=\frac{v}{u}$
$\therefore u=\frac{v}{m}=-40m$
Using lens formula,
$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}=\frac{1}{10}-\left(-\frac{1}{40}\right)$
$\therefore \frac{1}{f}=\frac{5}{40}$
$\Rightarrow f=8m.$
For plano-convex lens,
$\frac{1}{f}=(\mu -1)\left(\frac{1}{R}\right)$
$\therefore$ Radius of curvature $R=f(\mu -1)=8\left(\frac{4}{3}-1\right)$
$=2.67m$