how to find refractive index of aliquid using plane mirror and convex lens

First place the mirror and the convex lens as shown below. Then a needle is placed above the convex lens on a stand. The position of the needle is fixed in such a way that a distinct image of the needle is visible at the same position as that of the object (needle).

Measure the length PA. This is the focal length of the convex lens.

Now add some drop of the liquid of which refractive index is to be found between the lens and the mirror.

Again fix the needle such that the distinct image is formed at the same position as that of the object.

Measure QA. This is the focal length of the combination of the convex lens and the plano-concave lens formed by the liquid.

So,

Focal length of the lens combination is = F

Focal length of the convex lens is = F₁

Focal length of the liquid lens is = F₂

And,

1/F = 1/F₁ + 1/F₂

=> F₂ = (F₁F)/(F₁ + F)

Now, measure the curvature height of the convex lens using a spherometer, suppose it is ‘h’. And the distance between the spherometer legs is ‘a’.

So, the radius of curvature of the liquid lens is, R = a²/(6h)

Suppose, ‘n’ is the refractive index of the liquid.

Then,

n = 1 + R/F₂

We know R and F₂, so we have the refractive index of the liquid.