Fig. shows an equiconvex lens (of refractive index `1.5`) in contact with a liquid layer on top of a plane mirror. A small needle with its tip on the principal axis is moved along the axis until its inverted image is found at the position of the needle. The distance of the needle from the lens is measured to be `45.0 cm`. The liquid is removed and the experiment is repeated. The new distance is measured to be `30.0 cm`. What is the refractive index of the liquid ?

Explanation

Let focal length of convex lens of glass `= f_1 = 30 cm`,
focal length of plano concave lens of liquid `= f_2`
Combined focal length, `F = 45.0 cm`
As `(1)/(f_1)+(1)/(f_2)=(1)/(F) :. (1)/(f_2)=(1)/(F)-(1)/(f_1)=(1)/(45)-(1)/(30)= -(1)/(90)`
`f_2 = - 90 cm`
For glass lens, let `R_1 = R, R_2 = -R`
As `(1)/(f_1)=(mu - 1) ((1)/(R_1) - (1)/(R_2)) :. (1)/(30)=((3)/(2) - 1) ((1)/( R) + (1)/( R)) = (1)/(2) xx (2)/( R) = (1)/( R)`
`R = 30 cm`
For liquid lens, `R_1 = -R = -30.0 cm, R_2 = oo`
`(1)/(f_2)=(mu - 1) ((1)/(R_1) - (1)/(R_2)) = (mu - 1) ((1)/(-30) - (1)/(oo))`
or `(1)/(-90) = (mu - 1) xx (1)/(-30)`
`mu_l - 1 = (30)/(90) = (1)/(3)`
`mu_l = 1 + (1)/(3) = (4)/(3)`.