Question

An object when placed in front of a plano-convex lens forms a real image at a distance of $10 \,m$ 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

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