Question

To find the refractive indexes of (a)water (b) oil using a plane mirror, an equiconvex lens, and an adjustable object needle.

Answer 1

we have to find the refractive indexes of (a)water (b) oil using a plane mirror, an equiconvex lens, and an adjustable object needle.

$\textbf{need apparatus}$ : A plane mirror, water, oil , damp stand , an optical needle , a convex lens , plumb line , knitting needle , a metre scale , glass slab and spherometer.

if we assume that, $f_1$ denotes the focal length of convex lens , $f_2$ denotes the focal length of liquid lens and f denotes the focal length of their combination
then, $\frac{1}{f_1}+\frac{1}{f_2}=\frac{1}{f}$

so, $\bf{f_2=\frac{ff_1}{f-f_1}}$

actually, liquid lens is formed a plano -convex lens with $R_1=R,R_2=\infty$then using lens maker formula,
$\frac{1}{f_2}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$
$\frac{1}{f_2}=\frac{(\mu-1)}{R}$

$\frac{R}{f_2}=(\mu-1)$

$\bf{\mu=\frac{R}{f_2}+1}$

here R is the radius of the curvature of convex lens
the radius of the lower surface of the convex lens is given by, $\bf{R=\frac{l^2}{6h}+\frac{h}{2}}$

where l is the average distance between legs of spherometer and h is the difference in the reading of spherometer when placed first on the convex lens and then on plane mirror .

from above mentioned formulae we can get all required terms.