A symmetric biconvex lens of radius of curvature R and made of glass of refractive index 1.5, is placed on a layer of liquid placed on top of a plane mirror as shown in the figure. An optical needle with its tip on the principal axis of the lens is moved along the axis until its real, inverted image coincides with the needle itself. The distance of the needle from the lens is measured to be x. On removing the liquid layer and repeating the experiment, the distance is found to be y. Obtain the expression for the refractive index of the liquid in terms of x and y.
Given
Focal length of convex lens f1 = y
Refractive index of lens μ1 = 1.5
Let focal length of liquid be f2
And refractive index of liquid be μ2.
Let the focal length of system (lens + liquid) be f = x(given)
Now equivalent focal length is given as
Let radius of curvature of lens be R.
Then for a convex lens R1 = R and R2 = -R
From lens maker formula
Substituting the value of f1 and μ1, to find R
Since liquid acts as a plane mirror, therefore, R1 = R = ycm and R2 = ∞.
Applying lens maker formula for liquid, we get
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