An equiconvex lens with radii of curvature of magnitude R each is put over a liquid layer poured on top of a plane mirror. A small needle, with its tip on the principal axis of the lens, is moved along the axis until its inverted real image coincides with the needle itself. The distance of the needle from the lens is measured to be 'a'. On removing the liquid layer and repeating the experiment the distance is found to be 'b'.
Given that the two values of distances measured represent the focal length values in the two cases, obtain a formula for the refractive index of the liquid.

Explanation

Here, combined focal length of glass lens and liquid lens, `F = a`, and Focal length of convex lens, `f_1 = b` If `f_2` is focal length of liquid lens, then
`(1)/(f_1)+(1)/(f_2)=(1)/(F)`
`(1)/(f_2)=(1)/(F)-(1)/(f_1) = ((1)/(a) - (1)/(b))`
The liquid lens is plano concave lens for which `R_1 = -r, R_2 = oo`
From `(1)/(f_2)=(mu -1) ((1)/(R_1) - (1)/(R_2))`
`((1)/(a) - (1)/(b)) = (mu - 1) ((1)/(-r) - (1)/(- oo))`
`:. (mu - 1) = (r)/(b) - (r)/(a)`
`mu = 1 + (r)/(b) - (r)/(a)`.