Question

The image of an object, formed by a plano-convex lens at a distance of $8m$ behind the lens, is real and is one-third the size of the object. The wavelength of light inside the lens is $\frac{2}{3}$ times the wavelength in free space. The radius of the curved surface of the lens is:

• A. $1m$
• B. $2m$
• C. $3m$
• D. $4m$

Answer 1

The correct option is C $3m$

$\lambda$ inside the lens $=\frac{2}{3}\lambda$ in free space

$\Rightarrow \frac{\mu insidethelens}{\mu freespace}=\frac{\lambda freespace}{\lambda insidelens}=\frac{3}{2}$

$\Rightarrow \mu lens=\frac{3}{2}$

$\Rightarrow \frac{1}{f}=\left(\frac{1-\mu }{-R}\right)=\left(\frac{0.5}{R}\right)$

$\Rightarrow f=2R$

Given : $h_i=\frac{1}{3}{h}_0$ and $v=8m$

By lens formula $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$

And $m=\frac{v}{u}=\left(\frac{-1}{3}\right)$

$\Rightarrow \frac{8}{-x}=\frac{-1}{3}$

$\Rightarrow x=24m$

Now putting $v=8m$ and $u=-24$ in lens formula

$\Rightarrow \frac{1}{f}=\frac{1}{8}-\frac{1}{(-24)}=\frac{1}{6}$

$\Rightarrow f=6m.$

So, $f=2R=6m$

$\Rightarrow R=3m$ (C).

Hence, the answer is $3m.$