Question

On a horizontal plane mirror, a thin equiconvex lens of glass is placed and when the space between the lens and mirror is filled with a liquid, an object held at a distance D which is 30cm vertically above the lens is found to coincide with its own image as shown in Figure. If equiconvex lens of glass has refractive index `mu=1.5` and radius of curvature `R=20cm` then find refractive index of the liquid.

Answer 1

If an object has to coincide with its image, then the ray have to retrace its path, hence, it is concluded that object is at the focus of the combination.
Focus of equivalent combination.
`(1)/(F)=(1)/(f_(g))+(1)/(f_(1))`
`(1)/(f_(g))=(1.5-1)(2)/(R)==(1)/(R)`
`(1)/(f_(1))=(mu-1)(-(1)/(R)-(1)/(oo))=-((mu-1))/(R)`
`(1)/(F)=(1)/(R)-((mu-1)/(R))=((2-mu)/(R))`
Since, `F=D`
`:. mu=2-(R)/(D)`